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Dear DrProf RogerJ-B WetsHere are the proofs of your article
bull You can submit your corrections online or by faxbull For online submission please insert your corrections in the online correction form Always
indicate the line number to which the correction refersbull For fax submission please ensure that your corrections are clearly legible Use a fine black
pen and write the correction in the margin not too close to the edge of the pagebull Together with the proof please return the cover sheet (including the Copyright Transfer
Statement) and the Offprint Order Form They can either be scanned and sent electronicallyor sent by fax
bull Remember to note the journal title article number and your name when sending your responsevia e-mail fax or regular mail
bull Check the metadata sheet to make sure that the header information especially author namesand the corresponding affiliations are correctly shown
bull Check the questions that may have arisen during copy editing and insert your answerscorrections
bull Check that the text is complete and that all figures tables and their legends are included Alsocheck the accuracy of special characters equations and electronic supplementary material ifapplicable If necessary refer to the Edited manuscript
bull The publication of inaccurate data such as dosages and units can have serious consequencesPlease take particular care that all such details are correct
bull Please do not make changes that involve only matters of style We have generally introducedforms that follow the journalrsquos styleSubstantial changes in content eg new results corrected values title and authorship are notallowed without the approval of the responsible editor In such a case please contact theEditorial Office and return hisher consent together with the proof
bull If we do not receive your corrections within 48 hours we will send you a reminder
Please noteYour article will be published Online First approximately one week after receipt of your correctedproofs This is the official first publication citable with the DOI Further changes are thereforenot possibleAfter online publication subscribers (personalinstitutional) to this journal will have access to thecomplete article via the DOI using the URL httpdxdoiorg[DOI]If you would like to know when your article has been published online take advantage of our freealert service For registration and further information go to httpwwwspringerlinkcomDue to the electronic nature of the procedure the manuscript and the original figures will only bereturned to you on special request When you return your corrections please inform us if you wouldlike to have these documents returnedThe printed version will follow in a forthcoming issue
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RogerJ-B Wets RogerJ-B WetsDepartment of MathematicsUniversity of CaliforniaDavis USA
Department of MathematicsUniversity of CaliforniaDavis USA
q q
Metadata of the article that will be visualized in OnlineFirst
ArticleTitle Variational convergence of bivariate functions lopsided convergenceArticle Sub-Title
Journal Name Mathematical Programming
Corresponding Author Family Name WetsParticle
Given Name Roger J-BSuffix
Division Department of Mathematics
Organization University of California
Address Davis USA
Email rjbwetsucdavisedu
Author Family Name JofreacuteParticle
Given Name AlejandroSuffix
Division Ingeneria Matematica
Organization Universidad de Chile
Address Santiago Chile
Email ajofredimuchilecl
Schedule
Received 11 September 2005
Revised
Accepted 3 April 2006
Abstract We explore convergence notions for bivariate functions that yield convergence and stability results for theirmaxinf (or minsup) points This lays the foundations for the study of the stability of solutions to variationalinequalities the solutions of inclusions of Nash equilibrium points of non-cooperative games and Walraseconomic equilibrium points of fixed points of solutions to inclusions the primal and dual solutions ofconvex optimization problems and of zero-sum games These applications will be dealt with in a couple ofaccompanying papers
Keywords (separated by -) Lopsided convergence - Maxinf-points - Ky Fan functions - Variational inequalities - Epi-convergence
Mathematics SubjectClassification (2000)(separated by -)
65K10 - 90C31 - 91A10 - 47J20 - 47J30 - 49J45
Footnote Information Dedicated to A Auslender in recognition of his valuable contributions to Mathematical Programmingfoundations and numerical proceduresResearch supported in part by grants of the National Science Foundation and Fondap-Matematicas AplicadasUniversidad de Chile
Journal 10107 Article 122 Dear Author During the process of typesetting your article the following queries have arisen Please check your typeset proof carefully against the queries listed below and mark the necessary changes either directly on the proofonline grid or in the lsquoAuthorrsquos responsersquo area provided below
Author Query Form
Query Details required Authorrsquos response 1 Please check the author name
ldquoRoger J -B Wetsrdquo is correctly tagged (given and family name)
unco
rrec
ted
proo
f
Math Program Ser BDOI 101007s10107-007-0122-8
FULL LENGTH PAPER
Variational convergence of bivariate functions lopsidedconvergence
Alejandro Jofreacute middot Roger J-B Wets
Received 11 September 2005 Accepted 3 April 2006copy Springer-Verlag 2007
Abstract We explore convergence notions for bivariate functions that yield1
convergence and stability results for their maxinf (or minsup) points This lays the2
foundations for the study of the stability of solutions to variational inequalities the3
solutions of inclusions of Nash equilibrium points of non-cooperative games and4
Walras economic equilibrium points of fixed points of solutions to inclusions the5
primal and dual solutions of convex optimization problems and of zero-sum games6
These applications will be dealt with in a couple of accompanying papers7
Keywords Lopsided convergence middot Maxinf-points middot Ky Fan functions middot Variational8
points of non-cooperative games equilibrium points of zero-sum games etc We make33
a distinction between the situations when the bivariate function is generated from a34
single-valued mapping [11] or when the mapping can also be set-valued [10]35
The major tool is the notion of lopsided convergence that was introduced in [2]36
but is modified here so that a wider class of applications can be handled The major37
adjustment is that bivariate functions are not as in [2] no longer defined on all of38
Rn timesR
m with values in the extended reals but are now only finite-valued on a specific39
product C times D with C D subsets of Rn and R
m Dealing with lsquogeneralrsquo bivariate40
functions defined on the full product space was in keeping with the elegant work41
of Rockafellar [13] on duality relations for convexndashconcave bivariate functions and42
the subsequent work [3] on the epihypo-convergence of saddle functions However43
our present analysis actually shows that notwithstanding its esthetic allurement one44
should not cast bivariate functions even in the convexndashconcave case in the general45
extended-real valued framework In some way this is in contradiction with the uni-46
variate case where the extension by allowing for the values plusmninfin of functions defined47
on a (constrained) set to all of Rn has been so effectively exploited to derive a lsquounifiedrsquo48
convergence and differentiation theory [514] We shall show that some of this can be49
recovered but one must first make a clear distinction between max-inf problems and50
min-sup ones and only then one can generate the appropriate extensions after all51
also in the univariate case one makes a clear distinction when extending a function in52
a minimization setting or a maximization setting53
In order to be consistent in our presentation and to set up the results required54
later on we begin by a presentation of the theory of epi-convergence for real-valued55
univariate functions that are only defined on a subset of Rn No lsquonewrsquo results are56
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
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Variational convergence of bivariate functions
actually derived although a revised formulation is required We make the connection57
with the standard approach ie when these (univariate) functions are extended real-58
valued We then turn to lopsided convergence and point out the shortcomings of an59
lsquoextended real-valuedrsquo approach Finally we exploit our convergence result to obtain60
a extension of Ky Fan inequality [7] to situations when the domain of definition of the61
bivariate function is not necessarily compact62
2 Epi-convergence63
One can always represent an optimization problem involving constraints or not as64
one of minimizing an extended real-valued function In the case of a constrained-65
minimization problem simply redefine the objective as taking on the value infin out-66
side the feasible region the set determined by the constraints In this framework the67
canonical problem can be formulated as one of minimizing on all of Rn an extended68
real-valued function f Rn rarr R Approximation issues can consequently be stud-69
ied in terms of the convergence of such functions This has lead to the notion of70
epi-convergence1 that plays a key role in lsquoVariational Analysisrsquo [1514] when deal-71
ing with a maximization problem it is hypo-convergence the convergence of the72
hypographs that is the appropriate convergence notion73
Henceforth we restrict our development to the lsquominimization settingrsquo but at the74
end of this section we translate results and observations to the lsquomaximizationrsquo case75
As already indicated in Variational Analysis one usually deals with76
fcn(Rn) = f R
n rarr R
77
the space of extended real-valued functions that are defined on all of Rn even allowing78
for the possibility that they are nowhere finite-valued Definitions properties limits79
etc generally do not refer to the domain on which they are finite For reasons that will80
become clearer when we deal with the convergence of bivariate functions we need to81
depart from this simple and very convenient paradigm Our focus will be on82
f v-fcn(Rn) = f D rarr R
∣∣ for some empty = D sub R
n83
the class of all finite-valued functions with non-empty domain D sub Rn It must be84
understood that in this notation Rn does not refer to the domain of definition but to85
the underlying space that contains the domains on which the functions are defined86
The epigraph of a function f is always the set of all points in Rn+1 that lie on87
or above the graph of f irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) If88
f D rarr R belongs to f v-fcn(Rn) then89
epi f = (x α) isin D times R
∣∣ α ge f (x)
sub Rn+190
1 For extensive references and a survey of the field one can consult [15] and in particular the Commentarysection that concludes [14 Chap 7]
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
and if f belongs to fcn(Rn) then91
epi f = (x α) isin R
n+1∣∣α ge f (x)
92
A function f is lsc (=lower semicontinuous) if its epigraph is closed as a subset of93
Rn+1 ie epi f = cl(epi f ) with cl denoting closure [14 Theorem 16]294
So when f isin f v-fcn(Rn) lsc implies3 that for all xν isin D rarr x 95
ndash if x isin D liminfν f (xν) ge f (x) and96
ndash if x isin cl D D f (xν) rarr infin97
In our lsquominimizationrsquo framework cl f denotes the function whose epigraph is the98
closure relative to Rn+1 of the epigraph of f ie the lsc-regularization of f Its99
possible that when f isin f v-fcn cl f might be defined on a set thats strictly larger than100
D but always contained in cl D101
Lets now turn to convergence issues Recall that set-convergence in the Painleveacutendash102
Kuratowski sense [14 Sect 4B] is defined as follows Cν rarr C sub Rn if103
ndash (a-set) all cluster points of a sequence
xν isin CννisinIN belong to C 104
ndash (b-set) for each x isin C one can find a sequence xν isin Cν rarr x 105
When just condition (a-set) holds then C is then the outer limit of the sequence106 Cν
νisinIN and when its just (b-set) that holds then C is the inner limit [14 Chap 4107
Sect 2] Note that whenever C is the limit the outer- or the inner-limit its closed108
[14 Proposition 44] and that C = empty if and only if the sequence Cν eventually lsquoes-109
capesrsquo from any bounded set [14 Corollary 411] Moreover if the sequenceCν
νisinIN110
consists of convex sets its inner limit and its limit if it exists are also convex [14111
Proposition 415]112
Definition 1 (epi-convergence) A sequence of functions
f ν ν isin IN whose113
domains lie in Rn epi-converges to a function f when epi f ν rarr epi f as subsets114
of Rn+1 again irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) One then writes115
f ν rarre f 116
Figure 1 provides an example of two functions f and f ν that are close to each117
other in terms of the distance between their epigraphsmdashie the distance between the118
location of the two jumpsmdashbut are pretty far from each other pointwise or with respect119
to the infin-normmdashie the size of the jumps120
Let f ννisinIN be a sequence of functions with domains in Rn When121
ndash epi f is the outer limit of epi f ννisinIN one refers to f as the lower epi-limit of the122
functions f ν 123
ndash epi f is the inner limit of the epi f ν one refers to f as the upper epi-limit of the124
functions f ν 125
Of course f is the epi-limit of the sequence if its both the lower and upper epi-limit126
2 Throughout its implicitly assumed that Rn is equipped with its usual Euclidiean topology
3 Indeed if liminfν f (xν) lt infin then for some subsequence νk f (xνk ) rarr α isin R and because epi f isclosed it implies that (x α) isin epi f which would place x in the domain of f contradicting x isin D
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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proo
f
Variational convergence of bivariate functions
x
f
f
Fig 1 f and f ν epigraphically close to each other
Proposition 1 (properties of epi-limits) Let f ννisinIN be a sequence of functions with127
domains in Rn Then the lower and upper epi-limits and the epi-limit if it exists are128
all lsc Moreover if the functions f ν are convex so is the upper epi-limit and the129
epi-limit if it exists130
Proof Follows immediately from the properties of set-limits 131
The last proposition implies in particular that the family of lsc functions is closed132
under epi-convergence133
The definition of epi-convergence for families of functions in fcn(Rn) is the usual134
one [14 Chap 7 Sect B] with all the implications concerning the convergence of135
the minimizers and infimal values [14 Chap 7 Sect E] But in a certain sense the136
definition is lsquonewrsquo when the focus is on epi-convergent families in f v-fcn(Rn) and137
its for this class of functions that we need to know the conditions under which one can138
claim convergence of the minimizers and infimums We chose to make the presentation139
self-contained although as will be shown later one could also embed f v-fcn(Rn) in140
a subclass of fcn(Rn) and then appeal to the lsquostandardrsquo results but unfortunately this141
requires that the non-initiated reader plows through a substantial amount of material142
When f is an epi-limit its necessarily a lsc function since its epigraph is the set-143
limit of a collection of sets in Rn+1 Its epigraph is closed but its domain D is not144
necessarily closed Simply think of the collection of functions f ν = f for all ν with145
D = (0infin) and f (x) = 1x on D This collection clearly epi-converges to the lsc146
function f on D with closed epigraph but not with closed domain147
Lemma 1 (epi-limit value at boundary points) Suppose f D rarr R is the epi-limit148
of a sequence
f ν Dν rarr RνisinIN with all functions in f v-fcn(Rn) Then for any149
sequence xν isin Dν rarr x liminfν f ν(xν) gt minusinfin150
Proof We proceed by contradiction Suppose that xν isin Dν rarr x and151
liminfν f ν(xν) = minusinfin By assumption f gt minusinfin on D thus the xν cannot converge152
to a point in D ie necessarily x isin D If thats the case and since epi f ν rarr epi f 153
the line x times R would have to lie in epi f contradicting the assumption that f the154
epi-limit of the f ν belongs to f v-fcn(Rn) 155
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
Example 1 [an epi-limit thats not in f v-fcn(Rn)] Consider the sequence of functions156 f ν [0infin) rarr R
νisinIN with157
f ν(x) =
⎧⎪⎨
⎪⎩
minusν2x if 0 le x le νminus1
ν2x minus 2ν if νminus1 le x le 2νminus1
0 for x ge 2νminus1
158
Detail The functions f ν isin f v-fcn(R) and for the sequence xν = νminus1 f ν(xν) rarr minusinfin159
and f ν rarre f where f [0infin) rarr R with f equiv 0 on (0infin) and f (0) = minusinfin Thus the160
functions f ν epi-converge to f as functions in fcn(R) provided they are appropriately161
extended ie taking on the value infin on (minusinfin 0) But they do not epi-converge to a162
function in f v-fcn(R) 163
In addition to the lsquogeometricrsquo definition the next proposition provides an lsquoanalyticrsquo164
characterization of epi-converging sequences in f v-fcn(Rn)165
Proposition 2 [epi-convergence in f v-fcn(Rn)] Let
f D rarr R f ν Dν rarr166
R ν isin IN
be a collection of functions in f v-fcn(Rn) Then f ν rarre f if and only the167
following conditions are satisfied168
(a) forall xν isin Dν rarr x in D liminfν f ν(xν) ge f (x)169
(ainfin) for all xν isin Dν rarr x isin D f ν(xν)infin4170
(b) forallx isin D exist xν isin Dν rarr x such that limsupν f ν(xν) le f (x)171
Proof If epi f ν rarr epi f and xν isin Dν rarr x either lim infν f ν(xν) = α lt infin or172
not Lemma 1 reminds us that α = minusinfin is not a possibility In the first instance173
(x α) is a cluster point of(xν f ν(xν)) isin epi f ν
νisinIN and thus belongs to epi f 174
ie f (x) le α and hence (a) holds α gt minusinfin since otherwise f would not be finite175
valued on D If α = infin that means that f ν(xν)infin and x cannot belong to D176
and thus (ainfin) holds On the other hand if x isin D and thus f (x) is finite there is177
a(xν αν) isin epi f ν
νisinIN such that xν isin Dν rarr x isin D and αν rarr f (x) with178
αν ge f ν(xν) ie lim supν f ν(xν) le f (x) ie (b) is also satisfied179
Conversely if (a) and (ainfin) hold and (xν αν) isin epi f ν rarr (x α) then either x isin D180
or not recall also that in view of Lemma 1 α cannot be minusinfin since we are dealing181
with epi-convergence in f v-fcn(Rn) In the latter instance by (ainfin) α = infin so we are182
not dealing with a converging sequence of points (in Rn+1) and there is no need to183
consider this situation any further When x isin D since then lim infν f ν(xν) ge f (x)184
and αν ge f ν(xν) one has α ge f (x) and consequently (x α) belongs to epi f 185
this means that condition (a-set) is satisfied If (x α) isin epi f from (b) follows the186
existence of a sequence xν isin Dν rarr x such that lim supν f ν(xν) le f (x) le α We187
can then choose the αν ge f ν(xν) so that αν rarr α that yields (b-set) the second188
condition for the set-convergence of epi f ν rarr epi f 189
Theorem 1 (epi-convergence basic properties) Consider a sequence
f ν Dν rarr190
R ν isin IN sub f v-fcn(Rn) epi-converging to f D rarr R also in f v-fcn(Rn) Then191
4 means non-decreasing and converging to ie not necessarily monotonically
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
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Variational convergence of bivariate functions
lim supνrarrinfin
(inf f ν) le inf f192
Moreover if xk isin argminDνk f νk for some subsequence νk and xk rarr x then193
x isin argminD f and minDνk f νk rarr minD f 194
If argminD f is a singleton then every convergent subsequence of minimizers con-195
verges to argminD f 196
Proof Let xlinfinl=1 be a sequence in D such that f (xl) rarr inf f By 2(b) for each l197
one can find a sequence xνl isin Dν rarr xl such that lim supν f ν(xνl) le f (xl) Since198
for all ν inf f ν le f ν(xνl) it follows that for all l199
lim supν
(inf f ν) le lim supν
f ν(xνl) le f (xl)200
and one has lim supν(inf f ν) le inf f since f (xl) rarr inf f 201
For the sequence xk isin Dνk rarr x from the above and 2(a)202
inf f ge lim supk
f νk (xk) ge lim infk
f νk (xk) ge f (x)203
ie x minimizes f on D and f νk (xk) = minDνk f νk rarr minD f 204
Finally since every convergent subsequence of minimizers of the functions f ν205
converges to a minimizer of f it follows that it must converge to the unique minimizer206
when argminD f is a singleton 207
In most of the applications we shall rely on a somewhat more restrictive notion208
than lsquosimplersquo epi-convergence to guarantee the convergence of the infimums209
Definition 2 (tight epi-convergence) The sequence210
f ν Dν rarr RνisinIN sub f v-fcn(Rn) epi-converges tightly to f D rarr R isin211
f v-fcn(Rn) if f ν rarre f and for all ε gt 0 there exist a compact set Bε and an in-212
dex νε such that213
forall ν ge νε inf BεcapDν f ν le inf Dν f ν + ε214
Theorem 2 (convergence of the infimums) Let f ν Dν rarr RνisinIN sub f v-fcn(Rn)215
be a sequence of functions that epi-converges to the function f D rarr R also in216
f v-fcn(Rn) with inf D f finite Then they epi-converge tightly217
(a) if and only if inf Dν f ν rarr inf D f 218
(b) if and only if there exists a sequence εν 0 such that εν-argmin f ν rarr argmin f 219
Proof Lets start with necessity in (a) For given ε gt 0 the assumptions and Theorem 1220
imply221
lim infν
( infDνcapBε
f ν) le lim infν
(infDν
f ν) + ε le lim supν
(infDν
f ν) + ε le infD
f + ε lt infin222
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223
follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224
point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225
with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226
it follows that there is no such sequences with κ arbitrarily negative In other words227
excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228
from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229
a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230
Hence231
inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232
In combination with our first string of inequalities and the fact that ε gt 0 can be233
chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234
Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235
enough given any δ gt 0 to exhibit a compact set B such that lim supν
(inf BcapDν f ν
) le236
inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237
f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238
Let B be any compact set large enough to contain all the points xν Then inf B f ν le239
f ν(xν) for all ν so B has the desired property240
We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241
tion and consequently for ν large enough also αν is finite Since convergence of242
the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243
find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244
εν = αν minus αν 245
For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246
f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247
because f ν rarre f one obtains248
inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249
le lim infν(inf f ν) le limsupν(inf f ν) le inf f250
where the last inequality comes from Theorem 1 251
Remark 1 (convergence of domains) Although epi-convergence essentially implies252
convergence of the level sets [14 Proposition 77] it does not follow that it implies253
the convergence of their (effective) domains Indeed consider the following sequence254
f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255
function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256
This vigorously argues against the temptation of involving the convergence of their257
domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258
This concludes the presentation of the results that will be used in the sequel As259
indicated earlier its also possible to derive these results from those for extended real-260
valued functions To do so one identifies f v-fcn(Rn) with261
pr -fcn(Rn) = f isin fcn(Rn)
∣∣ minus infin lt f equiv infin
262
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
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ted
proo
f
Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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ted
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A Jofreacute R J -B Wets
η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
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ted
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f
Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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ted
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For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
I Permission to publishDear Springer Correction TeamI have checked the proofs of my article andq I have no corrections The article is ready to be published without changes
q I have a few corrections I am enclosing the following pagesq I have made many corrections Enclosed is the complete article
II Offprint orderq Offprint order enclosed q I do not wish to order offprintsRemarks
Date signature ______________________________________________________________________________
III Copyright Transfer Statement (sign only if not submitted previously)The copyright to this article is transferred to Springer-Verlag (respective to owner if other than Springer and for USgovernment employees to the extent transferable) effective if and when the article is accepted for publication Theauthor warrants that hisher contribution is original and that heshe has full power to make this grant The author signsfor and accepts responsibility for releasing this material on behalf of any and all co-authors The copyright transfercovers the exclusive right to reproduce and distribute the article including reprints translations photographicreproductions microform electronic form (offline online) or any other reproductions of similar nature
An author may self-archive an author-created version of hisher article on hisher own website and hisherinstitutionrsquos repository including hisher final version however heshe may not use the publisherrsquos PDF versionwhich is posted on httpwwwspringerlinkcom Furthermore the author may only post hisher version providedacknowledgement is given to the original source of publication and a link is inserted to the published article onSpringerrsquos website The link must be accompanied by the following text ldquoThe original publication is available athttpwwwspringerlinkcomrdquo
The author is requested to use the appropriate DOI for the article (go to the Linking Options in the article then toOpenURL and use the link with the DOI) Articles disseminated via httpwwwspringerlinkcom are indexedabstracted and referenced by many abstracting and information services bibliographic networks subscriptionagencies library networks and consortia
After submission of this agreement signed by the corresponding author changes of authorship or in the order ofthe authors listed will not be accepted by Springer
Date Authorrsquos signature ______________________________________________________________________
Offprint Order FormTo determine if your journal provides free offprintsplease check the journals instructions to authorsIf you do not return this order form we assume that you donot wish to order offprints
If you order offprints after the issue has gone topress costs are much higher Therefore we cansupply offprints only in quantities of 300 ormore after this timeFor orders involving more than 500 copiesplease ask the production editor for a quotation
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RogerJ-B Wets RogerJ-B WetsDepartment of MathematicsUniversity of CaliforniaDavis USA
Department of MathematicsUniversity of CaliforniaDavis USA
q q
Metadata of the article that will be visualized in OnlineFirst
ArticleTitle Variational convergence of bivariate functions lopsided convergenceArticle Sub-Title
Journal Name Mathematical Programming
Corresponding Author Family Name WetsParticle
Given Name Roger J-BSuffix
Division Department of Mathematics
Organization University of California
Address Davis USA
Email rjbwetsucdavisedu
Author Family Name JofreacuteParticle
Given Name AlejandroSuffix
Division Ingeneria Matematica
Organization Universidad de Chile
Address Santiago Chile
Email ajofredimuchilecl
Schedule
Received 11 September 2005
Revised
Accepted 3 April 2006
Abstract We explore convergence notions for bivariate functions that yield convergence and stability results for theirmaxinf (or minsup) points This lays the foundations for the study of the stability of solutions to variationalinequalities the solutions of inclusions of Nash equilibrium points of non-cooperative games and Walraseconomic equilibrium points of fixed points of solutions to inclusions the primal and dual solutions ofconvex optimization problems and of zero-sum games These applications will be dealt with in a couple ofaccompanying papers
Keywords (separated by -) Lopsided convergence - Maxinf-points - Ky Fan functions - Variational inequalities - Epi-convergence
Mathematics SubjectClassification (2000)(separated by -)
65K10 - 90C31 - 91A10 - 47J20 - 47J30 - 49J45
Footnote Information Dedicated to A Auslender in recognition of his valuable contributions to Mathematical Programmingfoundations and numerical proceduresResearch supported in part by grants of the National Science Foundation and Fondap-Matematicas AplicadasUniversidad de Chile
Journal 10107 Article 122 Dear Author During the process of typesetting your article the following queries have arisen Please check your typeset proof carefully against the queries listed below and mark the necessary changes either directly on the proofonline grid or in the lsquoAuthorrsquos responsersquo area provided below
Author Query Form
Query Details required Authorrsquos response 1 Please check the author name
ldquoRoger J -B Wetsrdquo is correctly tagged (given and family name)
unco
rrec
ted
proo
f
Math Program Ser BDOI 101007s10107-007-0122-8
FULL LENGTH PAPER
Variational convergence of bivariate functions lopsidedconvergence
Alejandro Jofreacute middot Roger J-B Wets
Received 11 September 2005 Accepted 3 April 2006copy Springer-Verlag 2007
Abstract We explore convergence notions for bivariate functions that yield1
convergence and stability results for their maxinf (or minsup) points This lays the2
foundations for the study of the stability of solutions to variational inequalities the3
solutions of inclusions of Nash equilibrium points of non-cooperative games and4
Walras economic equilibrium points of fixed points of solutions to inclusions the5
primal and dual solutions of convex optimization problems and of zero-sum games6
These applications will be dealt with in a couple of accompanying papers7
Keywords Lopsided convergence middot Maxinf-points middot Ky Fan functions middot Variational8
points of non-cooperative games equilibrium points of zero-sum games etc We make33
a distinction between the situations when the bivariate function is generated from a34
single-valued mapping [11] or when the mapping can also be set-valued [10]35
The major tool is the notion of lopsided convergence that was introduced in [2]36
but is modified here so that a wider class of applications can be handled The major37
adjustment is that bivariate functions are not as in [2] no longer defined on all of38
Rn timesR
m with values in the extended reals but are now only finite-valued on a specific39
product C times D with C D subsets of Rn and R
m Dealing with lsquogeneralrsquo bivariate40
functions defined on the full product space was in keeping with the elegant work41
of Rockafellar [13] on duality relations for convexndashconcave bivariate functions and42
the subsequent work [3] on the epihypo-convergence of saddle functions However43
our present analysis actually shows that notwithstanding its esthetic allurement one44
should not cast bivariate functions even in the convexndashconcave case in the general45
extended-real valued framework In some way this is in contradiction with the uni-46
variate case where the extension by allowing for the values plusmninfin of functions defined47
on a (constrained) set to all of Rn has been so effectively exploited to derive a lsquounifiedrsquo48
convergence and differentiation theory [514] We shall show that some of this can be49
recovered but one must first make a clear distinction between max-inf problems and50
min-sup ones and only then one can generate the appropriate extensions after all51
also in the univariate case one makes a clear distinction when extending a function in52
a minimization setting or a maximization setting53
In order to be consistent in our presentation and to set up the results required54
later on we begin by a presentation of the theory of epi-convergence for real-valued55
univariate functions that are only defined on a subset of Rn No lsquonewrsquo results are56
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
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Variational convergence of bivariate functions
actually derived although a revised formulation is required We make the connection57
with the standard approach ie when these (univariate) functions are extended real-58
valued We then turn to lopsided convergence and point out the shortcomings of an59
lsquoextended real-valuedrsquo approach Finally we exploit our convergence result to obtain60
a extension of Ky Fan inequality [7] to situations when the domain of definition of the61
bivariate function is not necessarily compact62
2 Epi-convergence63
One can always represent an optimization problem involving constraints or not as64
one of minimizing an extended real-valued function In the case of a constrained-65
minimization problem simply redefine the objective as taking on the value infin out-66
side the feasible region the set determined by the constraints In this framework the67
canonical problem can be formulated as one of minimizing on all of Rn an extended68
real-valued function f Rn rarr R Approximation issues can consequently be stud-69
ied in terms of the convergence of such functions This has lead to the notion of70
epi-convergence1 that plays a key role in lsquoVariational Analysisrsquo [1514] when deal-71
ing with a maximization problem it is hypo-convergence the convergence of the72
hypographs that is the appropriate convergence notion73
Henceforth we restrict our development to the lsquominimization settingrsquo but at the74
end of this section we translate results and observations to the lsquomaximizationrsquo case75
As already indicated in Variational Analysis one usually deals with76
fcn(Rn) = f R
n rarr R
77
the space of extended real-valued functions that are defined on all of Rn even allowing78
for the possibility that they are nowhere finite-valued Definitions properties limits79
etc generally do not refer to the domain on which they are finite For reasons that will80
become clearer when we deal with the convergence of bivariate functions we need to81
depart from this simple and very convenient paradigm Our focus will be on82
f v-fcn(Rn) = f D rarr R
∣∣ for some empty = D sub R
n83
the class of all finite-valued functions with non-empty domain D sub Rn It must be84
understood that in this notation Rn does not refer to the domain of definition but to85
the underlying space that contains the domains on which the functions are defined86
The epigraph of a function f is always the set of all points in Rn+1 that lie on87
or above the graph of f irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) If88
f D rarr R belongs to f v-fcn(Rn) then89
epi f = (x α) isin D times R
∣∣ α ge f (x)
sub Rn+190
1 For extensive references and a survey of the field one can consult [15] and in particular the Commentarysection that concludes [14 Chap 7]
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
and if f belongs to fcn(Rn) then91
epi f = (x α) isin R
n+1∣∣α ge f (x)
92
A function f is lsc (=lower semicontinuous) if its epigraph is closed as a subset of93
Rn+1 ie epi f = cl(epi f ) with cl denoting closure [14 Theorem 16]294
So when f isin f v-fcn(Rn) lsc implies3 that for all xν isin D rarr x 95
ndash if x isin D liminfν f (xν) ge f (x) and96
ndash if x isin cl D D f (xν) rarr infin97
In our lsquominimizationrsquo framework cl f denotes the function whose epigraph is the98
closure relative to Rn+1 of the epigraph of f ie the lsc-regularization of f Its99
possible that when f isin f v-fcn cl f might be defined on a set thats strictly larger than100
D but always contained in cl D101
Lets now turn to convergence issues Recall that set-convergence in the Painleveacutendash102
Kuratowski sense [14 Sect 4B] is defined as follows Cν rarr C sub Rn if103
ndash (a-set) all cluster points of a sequence
xν isin CννisinIN belong to C 104
ndash (b-set) for each x isin C one can find a sequence xν isin Cν rarr x 105
When just condition (a-set) holds then C is then the outer limit of the sequence106 Cν
νisinIN and when its just (b-set) that holds then C is the inner limit [14 Chap 4107
Sect 2] Note that whenever C is the limit the outer- or the inner-limit its closed108
[14 Proposition 44] and that C = empty if and only if the sequence Cν eventually lsquoes-109
capesrsquo from any bounded set [14 Corollary 411] Moreover if the sequenceCν
νisinIN110
consists of convex sets its inner limit and its limit if it exists are also convex [14111
Proposition 415]112
Definition 1 (epi-convergence) A sequence of functions
f ν ν isin IN whose113
domains lie in Rn epi-converges to a function f when epi f ν rarr epi f as subsets114
of Rn+1 again irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) One then writes115
f ν rarre f 116
Figure 1 provides an example of two functions f and f ν that are close to each117
other in terms of the distance between their epigraphsmdashie the distance between the118
location of the two jumpsmdashbut are pretty far from each other pointwise or with respect119
to the infin-normmdashie the size of the jumps120
Let f ννisinIN be a sequence of functions with domains in Rn When121
ndash epi f is the outer limit of epi f ννisinIN one refers to f as the lower epi-limit of the122
functions f ν 123
ndash epi f is the inner limit of the epi f ν one refers to f as the upper epi-limit of the124
functions f ν 125
Of course f is the epi-limit of the sequence if its both the lower and upper epi-limit126
2 Throughout its implicitly assumed that Rn is equipped with its usual Euclidiean topology
3 Indeed if liminfν f (xν) lt infin then for some subsequence νk f (xνk ) rarr α isin R and because epi f isclosed it implies that (x α) isin epi f which would place x in the domain of f contradicting x isin D
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functions
x
f
f
Fig 1 f and f ν epigraphically close to each other
Proposition 1 (properties of epi-limits) Let f ννisinIN be a sequence of functions with127
domains in Rn Then the lower and upper epi-limits and the epi-limit if it exists are128
all lsc Moreover if the functions f ν are convex so is the upper epi-limit and the129
epi-limit if it exists130
Proof Follows immediately from the properties of set-limits 131
The last proposition implies in particular that the family of lsc functions is closed132
under epi-convergence133
The definition of epi-convergence for families of functions in fcn(Rn) is the usual134
one [14 Chap 7 Sect B] with all the implications concerning the convergence of135
the minimizers and infimal values [14 Chap 7 Sect E] But in a certain sense the136
definition is lsquonewrsquo when the focus is on epi-convergent families in f v-fcn(Rn) and137
its for this class of functions that we need to know the conditions under which one can138
claim convergence of the minimizers and infimums We chose to make the presentation139
self-contained although as will be shown later one could also embed f v-fcn(Rn) in140
a subclass of fcn(Rn) and then appeal to the lsquostandardrsquo results but unfortunately this141
requires that the non-initiated reader plows through a substantial amount of material142
When f is an epi-limit its necessarily a lsc function since its epigraph is the set-143
limit of a collection of sets in Rn+1 Its epigraph is closed but its domain D is not144
necessarily closed Simply think of the collection of functions f ν = f for all ν with145
D = (0infin) and f (x) = 1x on D This collection clearly epi-converges to the lsc146
function f on D with closed epigraph but not with closed domain147
Lemma 1 (epi-limit value at boundary points) Suppose f D rarr R is the epi-limit148
of a sequence
f ν Dν rarr RνisinIN with all functions in f v-fcn(Rn) Then for any149
sequence xν isin Dν rarr x liminfν f ν(xν) gt minusinfin150
Proof We proceed by contradiction Suppose that xν isin Dν rarr x and151
liminfν f ν(xν) = minusinfin By assumption f gt minusinfin on D thus the xν cannot converge152
to a point in D ie necessarily x isin D If thats the case and since epi f ν rarr epi f 153
the line x times R would have to lie in epi f contradicting the assumption that f the154
epi-limit of the f ν belongs to f v-fcn(Rn) 155
123
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Example 1 [an epi-limit thats not in f v-fcn(Rn)] Consider the sequence of functions156 f ν [0infin) rarr R
νisinIN with157
f ν(x) =
⎧⎪⎨
⎪⎩
minusν2x if 0 le x le νminus1
ν2x minus 2ν if νminus1 le x le 2νminus1
0 for x ge 2νminus1
158
Detail The functions f ν isin f v-fcn(R) and for the sequence xν = νminus1 f ν(xν) rarr minusinfin159
and f ν rarre f where f [0infin) rarr R with f equiv 0 on (0infin) and f (0) = minusinfin Thus the160
functions f ν epi-converge to f as functions in fcn(R) provided they are appropriately161
extended ie taking on the value infin on (minusinfin 0) But they do not epi-converge to a162
function in f v-fcn(R) 163
In addition to the lsquogeometricrsquo definition the next proposition provides an lsquoanalyticrsquo164
characterization of epi-converging sequences in f v-fcn(Rn)165
Proposition 2 [epi-convergence in f v-fcn(Rn)] Let
f D rarr R f ν Dν rarr166
R ν isin IN
be a collection of functions in f v-fcn(Rn) Then f ν rarre f if and only the167
following conditions are satisfied168
(a) forall xν isin Dν rarr x in D liminfν f ν(xν) ge f (x)169
(ainfin) for all xν isin Dν rarr x isin D f ν(xν)infin4170
(b) forallx isin D exist xν isin Dν rarr x such that limsupν f ν(xν) le f (x)171
Proof If epi f ν rarr epi f and xν isin Dν rarr x either lim infν f ν(xν) = α lt infin or172
not Lemma 1 reminds us that α = minusinfin is not a possibility In the first instance173
(x α) is a cluster point of(xν f ν(xν)) isin epi f ν
νisinIN and thus belongs to epi f 174
ie f (x) le α and hence (a) holds α gt minusinfin since otherwise f would not be finite175
valued on D If α = infin that means that f ν(xν)infin and x cannot belong to D176
and thus (ainfin) holds On the other hand if x isin D and thus f (x) is finite there is177
a(xν αν) isin epi f ν
νisinIN such that xν isin Dν rarr x isin D and αν rarr f (x) with178
αν ge f ν(xν) ie lim supν f ν(xν) le f (x) ie (b) is also satisfied179
Conversely if (a) and (ainfin) hold and (xν αν) isin epi f ν rarr (x α) then either x isin D180
or not recall also that in view of Lemma 1 α cannot be minusinfin since we are dealing181
with epi-convergence in f v-fcn(Rn) In the latter instance by (ainfin) α = infin so we are182
not dealing with a converging sequence of points (in Rn+1) and there is no need to183
consider this situation any further When x isin D since then lim infν f ν(xν) ge f (x)184
and αν ge f ν(xν) one has α ge f (x) and consequently (x α) belongs to epi f 185
this means that condition (a-set) is satisfied If (x α) isin epi f from (b) follows the186
existence of a sequence xν isin Dν rarr x such that lim supν f ν(xν) le f (x) le α We187
can then choose the αν ge f ν(xν) so that αν rarr α that yields (b-set) the second188
condition for the set-convergence of epi f ν rarr epi f 189
Theorem 1 (epi-convergence basic properties) Consider a sequence
f ν Dν rarr190
R ν isin IN sub f v-fcn(Rn) epi-converging to f D rarr R also in f v-fcn(Rn) Then191
4 means non-decreasing and converging to ie not necessarily monotonically
123
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ted
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Variational convergence of bivariate functions
lim supνrarrinfin
(inf f ν) le inf f192
Moreover if xk isin argminDνk f νk for some subsequence νk and xk rarr x then193
x isin argminD f and minDνk f νk rarr minD f 194
If argminD f is a singleton then every convergent subsequence of minimizers con-195
verges to argminD f 196
Proof Let xlinfinl=1 be a sequence in D such that f (xl) rarr inf f By 2(b) for each l197
one can find a sequence xνl isin Dν rarr xl such that lim supν f ν(xνl) le f (xl) Since198
for all ν inf f ν le f ν(xνl) it follows that for all l199
lim supν
(inf f ν) le lim supν
f ν(xνl) le f (xl)200
and one has lim supν(inf f ν) le inf f since f (xl) rarr inf f 201
For the sequence xk isin Dνk rarr x from the above and 2(a)202
inf f ge lim supk
f νk (xk) ge lim infk
f νk (xk) ge f (x)203
ie x minimizes f on D and f νk (xk) = minDνk f νk rarr minD f 204
Finally since every convergent subsequence of minimizers of the functions f ν205
converges to a minimizer of f it follows that it must converge to the unique minimizer206
when argminD f is a singleton 207
In most of the applications we shall rely on a somewhat more restrictive notion208
than lsquosimplersquo epi-convergence to guarantee the convergence of the infimums209
Definition 2 (tight epi-convergence) The sequence210
f ν Dν rarr RνisinIN sub f v-fcn(Rn) epi-converges tightly to f D rarr R isin211
f v-fcn(Rn) if f ν rarre f and for all ε gt 0 there exist a compact set Bε and an in-212
dex νε such that213
forall ν ge νε inf BεcapDν f ν le inf Dν f ν + ε214
Theorem 2 (convergence of the infimums) Let f ν Dν rarr RνisinIN sub f v-fcn(Rn)215
be a sequence of functions that epi-converges to the function f D rarr R also in216
f v-fcn(Rn) with inf D f finite Then they epi-converge tightly217
(a) if and only if inf Dν f ν rarr inf D f 218
(b) if and only if there exists a sequence εν 0 such that εν-argmin f ν rarr argmin f 219
Proof Lets start with necessity in (a) For given ε gt 0 the assumptions and Theorem 1220
imply221
lim infν
( infDνcapBε
f ν) le lim infν
(infDν
f ν) + ε le lim supν
(infDν
f ν) + ε le infD
f + ε lt infin222
123
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If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223
follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224
point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225
with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226
it follows that there is no such sequences with κ arbitrarily negative In other words227
excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228
from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229
a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230
Hence231
inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232
In combination with our first string of inequalities and the fact that ε gt 0 can be233
chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234
Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235
enough given any δ gt 0 to exhibit a compact set B such that lim supν
(inf BcapDν f ν
) le236
inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237
f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238
Let B be any compact set large enough to contain all the points xν Then inf B f ν le239
f ν(xν) for all ν so B has the desired property240
We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241
tion and consequently for ν large enough also αν is finite Since convergence of242
the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243
find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244
εν = αν minus αν 245
For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246
f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247
because f ν rarre f one obtains248
inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249
le lim infν(inf f ν) le limsupν(inf f ν) le inf f250
where the last inequality comes from Theorem 1 251
Remark 1 (convergence of domains) Although epi-convergence essentially implies252
convergence of the level sets [14 Proposition 77] it does not follow that it implies253
the convergence of their (effective) domains Indeed consider the following sequence254
f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255
function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256
This vigorously argues against the temptation of involving the convergence of their257
domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258
This concludes the presentation of the results that will be used in the sequel As259
indicated earlier its also possible to derive these results from those for extended real-260
valued functions To do so one identifies f v-fcn(Rn) with261
pr -fcn(Rn) = f isin fcn(Rn)
∣∣ minus infin lt f equiv infin
262
123
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Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
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η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
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Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
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Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
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RogerJ-B Wets RogerJ-B WetsDepartment of MathematicsUniversity of CaliforniaDavis USA
Department of MathematicsUniversity of CaliforniaDavis USA
q q
Metadata of the article that will be visualized in OnlineFirst
ArticleTitle Variational convergence of bivariate functions lopsided convergenceArticle Sub-Title
Journal Name Mathematical Programming
Corresponding Author Family Name WetsParticle
Given Name Roger J-BSuffix
Division Department of Mathematics
Organization University of California
Address Davis USA
Email rjbwetsucdavisedu
Author Family Name JofreacuteParticle
Given Name AlejandroSuffix
Division Ingeneria Matematica
Organization Universidad de Chile
Address Santiago Chile
Email ajofredimuchilecl
Schedule
Received 11 September 2005
Revised
Accepted 3 April 2006
Abstract We explore convergence notions for bivariate functions that yield convergence and stability results for theirmaxinf (or minsup) points This lays the foundations for the study of the stability of solutions to variationalinequalities the solutions of inclusions of Nash equilibrium points of non-cooperative games and Walraseconomic equilibrium points of fixed points of solutions to inclusions the primal and dual solutions ofconvex optimization problems and of zero-sum games These applications will be dealt with in a couple ofaccompanying papers
Keywords (separated by -) Lopsided convergence - Maxinf-points - Ky Fan functions - Variational inequalities - Epi-convergence
Mathematics SubjectClassification (2000)(separated by -)
65K10 - 90C31 - 91A10 - 47J20 - 47J30 - 49J45
Footnote Information Dedicated to A Auslender in recognition of his valuable contributions to Mathematical Programmingfoundations and numerical proceduresResearch supported in part by grants of the National Science Foundation and Fondap-Matematicas AplicadasUniversidad de Chile
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ldquoRoger J -B Wetsrdquo is correctly tagged (given and family name)
unco
rrec
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Math Program Ser BDOI 101007s10107-007-0122-8
FULL LENGTH PAPER
Variational convergence of bivariate functions lopsidedconvergence
Alejandro Jofreacute middot Roger J-B Wets
Received 11 September 2005 Accepted 3 April 2006copy Springer-Verlag 2007
Abstract We explore convergence notions for bivariate functions that yield1
convergence and stability results for their maxinf (or minsup) points This lays the2
foundations for the study of the stability of solutions to variational inequalities the3
solutions of inclusions of Nash equilibrium points of non-cooperative games and4
Walras economic equilibrium points of fixed points of solutions to inclusions the5
primal and dual solutions of convex optimization problems and of zero-sum games6
These applications will be dealt with in a couple of accompanying papers7
Keywords Lopsided convergence middot Maxinf-points middot Ky Fan functions middot Variational8
points of non-cooperative games equilibrium points of zero-sum games etc We make33
a distinction between the situations when the bivariate function is generated from a34
single-valued mapping [11] or when the mapping can also be set-valued [10]35
The major tool is the notion of lopsided convergence that was introduced in [2]36
but is modified here so that a wider class of applications can be handled The major37
adjustment is that bivariate functions are not as in [2] no longer defined on all of38
Rn timesR
m with values in the extended reals but are now only finite-valued on a specific39
product C times D with C D subsets of Rn and R
m Dealing with lsquogeneralrsquo bivariate40
functions defined on the full product space was in keeping with the elegant work41
of Rockafellar [13] on duality relations for convexndashconcave bivariate functions and42
the subsequent work [3] on the epihypo-convergence of saddle functions However43
our present analysis actually shows that notwithstanding its esthetic allurement one44
should not cast bivariate functions even in the convexndashconcave case in the general45
extended-real valued framework In some way this is in contradiction with the uni-46
variate case where the extension by allowing for the values plusmninfin of functions defined47
on a (constrained) set to all of Rn has been so effectively exploited to derive a lsquounifiedrsquo48
convergence and differentiation theory [514] We shall show that some of this can be49
recovered but one must first make a clear distinction between max-inf problems and50
min-sup ones and only then one can generate the appropriate extensions after all51
also in the univariate case one makes a clear distinction when extending a function in52
a minimization setting or a maximization setting53
In order to be consistent in our presentation and to set up the results required54
later on we begin by a presentation of the theory of epi-convergence for real-valued55
univariate functions that are only defined on a subset of Rn No lsquonewrsquo results are56
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functions
actually derived although a revised formulation is required We make the connection57
with the standard approach ie when these (univariate) functions are extended real-58
valued We then turn to lopsided convergence and point out the shortcomings of an59
lsquoextended real-valuedrsquo approach Finally we exploit our convergence result to obtain60
a extension of Ky Fan inequality [7] to situations when the domain of definition of the61
bivariate function is not necessarily compact62
2 Epi-convergence63
One can always represent an optimization problem involving constraints or not as64
one of minimizing an extended real-valued function In the case of a constrained-65
minimization problem simply redefine the objective as taking on the value infin out-66
side the feasible region the set determined by the constraints In this framework the67
canonical problem can be formulated as one of minimizing on all of Rn an extended68
real-valued function f Rn rarr R Approximation issues can consequently be stud-69
ied in terms of the convergence of such functions This has lead to the notion of70
epi-convergence1 that plays a key role in lsquoVariational Analysisrsquo [1514] when deal-71
ing with a maximization problem it is hypo-convergence the convergence of the72
hypographs that is the appropriate convergence notion73
Henceforth we restrict our development to the lsquominimization settingrsquo but at the74
end of this section we translate results and observations to the lsquomaximizationrsquo case75
As already indicated in Variational Analysis one usually deals with76
fcn(Rn) = f R
n rarr R
77
the space of extended real-valued functions that are defined on all of Rn even allowing78
for the possibility that they are nowhere finite-valued Definitions properties limits79
etc generally do not refer to the domain on which they are finite For reasons that will80
become clearer when we deal with the convergence of bivariate functions we need to81
depart from this simple and very convenient paradigm Our focus will be on82
f v-fcn(Rn) = f D rarr R
∣∣ for some empty = D sub R
n83
the class of all finite-valued functions with non-empty domain D sub Rn It must be84
understood that in this notation Rn does not refer to the domain of definition but to85
the underlying space that contains the domains on which the functions are defined86
The epigraph of a function f is always the set of all points in Rn+1 that lie on87
or above the graph of f irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) If88
f D rarr R belongs to f v-fcn(Rn) then89
epi f = (x α) isin D times R
∣∣ α ge f (x)
sub Rn+190
1 For extensive references and a survey of the field one can consult [15] and in particular the Commentarysection that concludes [14 Chap 7]
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
and if f belongs to fcn(Rn) then91
epi f = (x α) isin R
n+1∣∣α ge f (x)
92
A function f is lsc (=lower semicontinuous) if its epigraph is closed as a subset of93
Rn+1 ie epi f = cl(epi f ) with cl denoting closure [14 Theorem 16]294
So when f isin f v-fcn(Rn) lsc implies3 that for all xν isin D rarr x 95
ndash if x isin D liminfν f (xν) ge f (x) and96
ndash if x isin cl D D f (xν) rarr infin97
In our lsquominimizationrsquo framework cl f denotes the function whose epigraph is the98
closure relative to Rn+1 of the epigraph of f ie the lsc-regularization of f Its99
possible that when f isin f v-fcn cl f might be defined on a set thats strictly larger than100
D but always contained in cl D101
Lets now turn to convergence issues Recall that set-convergence in the Painleveacutendash102
Kuratowski sense [14 Sect 4B] is defined as follows Cν rarr C sub Rn if103
ndash (a-set) all cluster points of a sequence
xν isin CννisinIN belong to C 104
ndash (b-set) for each x isin C one can find a sequence xν isin Cν rarr x 105
When just condition (a-set) holds then C is then the outer limit of the sequence106 Cν
νisinIN and when its just (b-set) that holds then C is the inner limit [14 Chap 4107
Sect 2] Note that whenever C is the limit the outer- or the inner-limit its closed108
[14 Proposition 44] and that C = empty if and only if the sequence Cν eventually lsquoes-109
capesrsquo from any bounded set [14 Corollary 411] Moreover if the sequenceCν
νisinIN110
consists of convex sets its inner limit and its limit if it exists are also convex [14111
Proposition 415]112
Definition 1 (epi-convergence) A sequence of functions
f ν ν isin IN whose113
domains lie in Rn epi-converges to a function f when epi f ν rarr epi f as subsets114
of Rn+1 again irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) One then writes115
f ν rarre f 116
Figure 1 provides an example of two functions f and f ν that are close to each117
other in terms of the distance between their epigraphsmdashie the distance between the118
location of the two jumpsmdashbut are pretty far from each other pointwise or with respect119
to the infin-normmdashie the size of the jumps120
Let f ννisinIN be a sequence of functions with domains in Rn When121
ndash epi f is the outer limit of epi f ννisinIN one refers to f as the lower epi-limit of the122
functions f ν 123
ndash epi f is the inner limit of the epi f ν one refers to f as the upper epi-limit of the124
functions f ν 125
Of course f is the epi-limit of the sequence if its both the lower and upper epi-limit126
2 Throughout its implicitly assumed that Rn is equipped with its usual Euclidiean topology
3 Indeed if liminfν f (xν) lt infin then for some subsequence νk f (xνk ) rarr α isin R and because epi f isclosed it implies that (x α) isin epi f which would place x in the domain of f contradicting x isin D
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functions
x
f
f
Fig 1 f and f ν epigraphically close to each other
Proposition 1 (properties of epi-limits) Let f ννisinIN be a sequence of functions with127
domains in Rn Then the lower and upper epi-limits and the epi-limit if it exists are128
all lsc Moreover if the functions f ν are convex so is the upper epi-limit and the129
epi-limit if it exists130
Proof Follows immediately from the properties of set-limits 131
The last proposition implies in particular that the family of lsc functions is closed132
under epi-convergence133
The definition of epi-convergence for families of functions in fcn(Rn) is the usual134
one [14 Chap 7 Sect B] with all the implications concerning the convergence of135
the minimizers and infimal values [14 Chap 7 Sect E] But in a certain sense the136
definition is lsquonewrsquo when the focus is on epi-convergent families in f v-fcn(Rn) and137
its for this class of functions that we need to know the conditions under which one can138
claim convergence of the minimizers and infimums We chose to make the presentation139
self-contained although as will be shown later one could also embed f v-fcn(Rn) in140
a subclass of fcn(Rn) and then appeal to the lsquostandardrsquo results but unfortunately this141
requires that the non-initiated reader plows through a substantial amount of material142
When f is an epi-limit its necessarily a lsc function since its epigraph is the set-143
limit of a collection of sets in Rn+1 Its epigraph is closed but its domain D is not144
necessarily closed Simply think of the collection of functions f ν = f for all ν with145
D = (0infin) and f (x) = 1x on D This collection clearly epi-converges to the lsc146
function f on D with closed epigraph but not with closed domain147
Lemma 1 (epi-limit value at boundary points) Suppose f D rarr R is the epi-limit148
of a sequence
f ν Dν rarr RνisinIN with all functions in f v-fcn(Rn) Then for any149
sequence xν isin Dν rarr x liminfν f ν(xν) gt minusinfin150
Proof We proceed by contradiction Suppose that xν isin Dν rarr x and151
liminfν f ν(xν) = minusinfin By assumption f gt minusinfin on D thus the xν cannot converge152
to a point in D ie necessarily x isin D If thats the case and since epi f ν rarr epi f 153
the line x times R would have to lie in epi f contradicting the assumption that f the154
epi-limit of the f ν belongs to f v-fcn(Rn) 155
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
Example 1 [an epi-limit thats not in f v-fcn(Rn)] Consider the sequence of functions156 f ν [0infin) rarr R
νisinIN with157
f ν(x) =
⎧⎪⎨
⎪⎩
minusν2x if 0 le x le νminus1
ν2x minus 2ν if νminus1 le x le 2νminus1
0 for x ge 2νminus1
158
Detail The functions f ν isin f v-fcn(R) and for the sequence xν = νminus1 f ν(xν) rarr minusinfin159
and f ν rarre f where f [0infin) rarr R with f equiv 0 on (0infin) and f (0) = minusinfin Thus the160
functions f ν epi-converge to f as functions in fcn(R) provided they are appropriately161
extended ie taking on the value infin on (minusinfin 0) But they do not epi-converge to a162
function in f v-fcn(R) 163
In addition to the lsquogeometricrsquo definition the next proposition provides an lsquoanalyticrsquo164
characterization of epi-converging sequences in f v-fcn(Rn)165
Proposition 2 [epi-convergence in f v-fcn(Rn)] Let
f D rarr R f ν Dν rarr166
R ν isin IN
be a collection of functions in f v-fcn(Rn) Then f ν rarre f if and only the167
following conditions are satisfied168
(a) forall xν isin Dν rarr x in D liminfν f ν(xν) ge f (x)169
(ainfin) for all xν isin Dν rarr x isin D f ν(xν)infin4170
(b) forallx isin D exist xν isin Dν rarr x such that limsupν f ν(xν) le f (x)171
Proof If epi f ν rarr epi f and xν isin Dν rarr x either lim infν f ν(xν) = α lt infin or172
not Lemma 1 reminds us that α = minusinfin is not a possibility In the first instance173
(x α) is a cluster point of(xν f ν(xν)) isin epi f ν
νisinIN and thus belongs to epi f 174
ie f (x) le α and hence (a) holds α gt minusinfin since otherwise f would not be finite175
valued on D If α = infin that means that f ν(xν)infin and x cannot belong to D176
and thus (ainfin) holds On the other hand if x isin D and thus f (x) is finite there is177
a(xν αν) isin epi f ν
νisinIN such that xν isin Dν rarr x isin D and αν rarr f (x) with178
αν ge f ν(xν) ie lim supν f ν(xν) le f (x) ie (b) is also satisfied179
Conversely if (a) and (ainfin) hold and (xν αν) isin epi f ν rarr (x α) then either x isin D180
or not recall also that in view of Lemma 1 α cannot be minusinfin since we are dealing181
with epi-convergence in f v-fcn(Rn) In the latter instance by (ainfin) α = infin so we are182
not dealing with a converging sequence of points (in Rn+1) and there is no need to183
consider this situation any further When x isin D since then lim infν f ν(xν) ge f (x)184
and αν ge f ν(xν) one has α ge f (x) and consequently (x α) belongs to epi f 185
this means that condition (a-set) is satisfied If (x α) isin epi f from (b) follows the186
existence of a sequence xν isin Dν rarr x such that lim supν f ν(xν) le f (x) le α We187
can then choose the αν ge f ν(xν) so that αν rarr α that yields (b-set) the second188
condition for the set-convergence of epi f ν rarr epi f 189
Theorem 1 (epi-convergence basic properties) Consider a sequence
f ν Dν rarr190
R ν isin IN sub f v-fcn(Rn) epi-converging to f D rarr R also in f v-fcn(Rn) Then191
4 means non-decreasing and converging to ie not necessarily monotonically
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functions
lim supνrarrinfin
(inf f ν) le inf f192
Moreover if xk isin argminDνk f νk for some subsequence νk and xk rarr x then193
x isin argminD f and minDνk f νk rarr minD f 194
If argminD f is a singleton then every convergent subsequence of minimizers con-195
verges to argminD f 196
Proof Let xlinfinl=1 be a sequence in D such that f (xl) rarr inf f By 2(b) for each l197
one can find a sequence xνl isin Dν rarr xl such that lim supν f ν(xνl) le f (xl) Since198
for all ν inf f ν le f ν(xνl) it follows that for all l199
lim supν
(inf f ν) le lim supν
f ν(xνl) le f (xl)200
and one has lim supν(inf f ν) le inf f since f (xl) rarr inf f 201
For the sequence xk isin Dνk rarr x from the above and 2(a)202
inf f ge lim supk
f νk (xk) ge lim infk
f νk (xk) ge f (x)203
ie x minimizes f on D and f νk (xk) = minDνk f νk rarr minD f 204
Finally since every convergent subsequence of minimizers of the functions f ν205
converges to a minimizer of f it follows that it must converge to the unique minimizer206
when argminD f is a singleton 207
In most of the applications we shall rely on a somewhat more restrictive notion208
than lsquosimplersquo epi-convergence to guarantee the convergence of the infimums209
Definition 2 (tight epi-convergence) The sequence210
f ν Dν rarr RνisinIN sub f v-fcn(Rn) epi-converges tightly to f D rarr R isin211
f v-fcn(Rn) if f ν rarre f and for all ε gt 0 there exist a compact set Bε and an in-212
dex νε such that213
forall ν ge νε inf BεcapDν f ν le inf Dν f ν + ε214
Theorem 2 (convergence of the infimums) Let f ν Dν rarr RνisinIN sub f v-fcn(Rn)215
be a sequence of functions that epi-converges to the function f D rarr R also in216
f v-fcn(Rn) with inf D f finite Then they epi-converge tightly217
(a) if and only if inf Dν f ν rarr inf D f 218
(b) if and only if there exists a sequence εν 0 such that εν-argmin f ν rarr argmin f 219
Proof Lets start with necessity in (a) For given ε gt 0 the assumptions and Theorem 1220
imply221
lim infν
( infDνcapBε
f ν) le lim infν
(infDν
f ν) + ε le lim supν
(infDν
f ν) + ε le infD
f + ε lt infin222
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223
follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224
point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225
with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226
it follows that there is no such sequences with κ arbitrarily negative In other words227
excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228
from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229
a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230
Hence231
inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232
In combination with our first string of inequalities and the fact that ε gt 0 can be233
chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234
Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235
enough given any δ gt 0 to exhibit a compact set B such that lim supν
(inf BcapDν f ν
) le236
inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237
f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238
Let B be any compact set large enough to contain all the points xν Then inf B f ν le239
f ν(xν) for all ν so B has the desired property240
We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241
tion and consequently for ν large enough also αν is finite Since convergence of242
the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243
find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244
εν = αν minus αν 245
For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246
f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247
because f ν rarre f one obtains248
inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249
le lim infν(inf f ν) le limsupν(inf f ν) le inf f250
where the last inequality comes from Theorem 1 251
Remark 1 (convergence of domains) Although epi-convergence essentially implies252
convergence of the level sets [14 Proposition 77] it does not follow that it implies253
the convergence of their (effective) domains Indeed consider the following sequence254
f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255
function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256
This vigorously argues against the temptation of involving the convergence of their257
domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258
This concludes the presentation of the results that will be used in the sequel As259
indicated earlier its also possible to derive these results from those for extended real-260
valued functions To do so one identifies f v-fcn(Rn) with261
pr -fcn(Rn) = f isin fcn(Rn)
∣∣ minus infin lt f equiv infin
262
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
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Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
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Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
the functions gν hypo-converge to g Maxinf-points for Fν and F are then maximizers599
of the corresponding functions gν and g The assertions now follow immediately600
from the convergence of the argmax of hypo-converging sequences cf Theorem [14601
Theorem 731] translated to the lsquomaximizationrsquo framework 602
To deal with a lsquominsuprsquo situations one can either repeat all the arguments changing603
inf to sup liminf to limsup and vice-versa or simply re-integrate the questions to604
the lsquomaxinfrsquo framework by changing signs of the approximating and limit bivariate605
functions606
123
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Variational convergence of bivariate functions
4 Ky Fanrsquos Inequality extended607
The class of usc functions is closed under hypo-converge [14 Theorem 74] and so608
is the class of concave usc functions [14 Theorem 717] A class of functions that609
is closed under lopsided convergence is the class of Ky Fan functions (Theorem 8)610
We exploit this result to obtain a generalization of Ky Fan Inequality that allows us611
to claim existence of maxinf-points in situations when the domain of definition of the612
Ky Fan function is not necessarily compact613
Definition 7 A bivariate function F C times D rarr R with Cand D convex sets in614
f v-biv(Rn+m) is called a Ky Fan function if615
(a) forall y isin D x rarr F(x y) is usc on C 616
(b) forall x isin C y rarr F(x y) is convex on D617
Note that the sets C or D are not required to be compact618
Theorem 8 (lop-limits of Ky Fan functions) The lopsided limit F C times D rarr R of a619
sequence
Fν Cν times Dν rarr RνisinIN of Ky Fan functions in f v-biv(Rn+m) is also a620
Ky Fan function621
Proof For the convexity of y rarr F(x y) let xν isin Cν rarr x isin C be the sequence set622
forth by 3(b) and y0 yλ y1 isin D with yλ = (1 minus λ)y0 + λy1 for λ isin [0 1] In view623
of 3(a) we can choose sequences
y0ν isin Dν rarr y0
y1ν isin Dν rarr y1
such that624
Fν(xν y0ν) rarr F(x y0) and Fν(xν y1ν) rarr F(x y1) Let yλν = (1 minus λ)y0ν +625
λy1ν yλν isin Dν since the functions Fν(x middot) are convex and the sequence yλννisinIN626
certainly converges to yλ For all ν one has627
Fν(xν yλν) le (1 minus λ)Fν(xν y0ν) + λFν(xν y1ν)628
Taking lininf on both sides yields629
F(x yλ) le liminfν Fν(xν yλν) le (1 minus λ)F(x y0) + λF(x y1)630
that establishes the convexity of F(x middot)631
To prove the upper semicontinuity of F with respect to x-variable we show that632
for y isin D633
hypo F(middot y) is the inner set-limit of the hypo Fν(middot yν)634
where the limit is with respect to all sequences yν isin DννisinIN converging to y and635
ν rarr infin This yields the upper semicontinuity since the inner set-limit is always636
closed and a function is usc if and only if its hypograph is closed We have to show637
that if (x α) isin hypo F(middot y) then whenever yν isin Dν rarr y one can find (xν αν) isin638
hypo Fν(middot yν) such that (xν αν) rarr (x α) But that follows immediately from 3(a)639
since we can adjust the αν le Fν(xν yν) so that they converge to α le F(x y) 640
Given a Ky Fan function with compact domain and non-negative on the diagonal641
we have the following important existence result642
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
Lemma 2 (Ky Fanrsquos Inequality [7] [4 Theorem 635]) Suppose F C times C rarr R is643
a Ky Fan function with C compact Then the set of maxinf-points of F is a nonempty644
subset of C Moreover if F(x x) ge 0 (on C times C) then for every maxinf-point x of645
C F(x middot) ge 0 on C646
One of the consequences of the lopsided convergence is an extension of the Ky Fanrsquos647
Inequality to the case when it is not possible to apply it directly because one of the648
conditions is not satisfied for example the compactness of the domain However we are649
able to approach the bivariate function F by a sequence FννisinIN defined on compact650
sets Cν This procedure could be useful in many situation where the original maxinf-651
problem is unbounded and then the problem is approached by a family of truncated652
maxinf-problems Such is the case for example when we consider as variables in653
the original problem the multipliers associated to inequality constraints or when the654
original problem is a Walras equilibrium with a positive orthant as consumption set655
in [8] one is precisely confronted with such situations Another simple illustrative656
example follows the statement of the theorem657
Theorem 9 (Extension of Ky Fanrsquos Inequality) Let empty = C sub Rn and F a finite-valued658
bivariate function defined on CtimesC Suppose one can find sequences of compact convex659
setsCν sub R
n
and (finite-valued) Ky Fan functions Fν Cν times Cν rarr RνisinIN lop-660
converging y-tightly to F then every cluster point x of any sequence xν ν isin IN of661
maxinf-points of the Fν is a maxinf-point of F662
Proof Ky Fanrsquos Inequality 2 implies that for all ν the set of maxinf-points of Fν is663
non-empty On the other hand in view of Theorems 8 and 4 any cluster point of such664
maxinf-points will be a maxinf-point of F 665
Example 2 (Extended Ky Fanrsquos Inequality applied) We consider a Ky Fan function666
F(x y) = sin x + (y + 1)minus1 defined on the set [0infin)2 Although667
inf yisin[0infin) F(x y) = sin x668
and the set maxinf-points is not empty we cannot apply Ky Fan Inequality because669
the domain of F is not compact the function F(middot y) is not even sup-compact670
Detail If we consider the functions Fν(x y) = sin x + (y + 1)minus1 on the compact671
domains [0 ν]2 one can apply Ky Fanrsquos Inequality Indeed in this case we have672
inf yisin[0ν) F(x y) = sin x + (ν + 1)minus1673
that converges pointwise and hypo- to sin x and674
argmaxyisin[0ν] inf yisin[0ν] F(x y) = π2 + 2kπ
∣∣ k isin IN
675
Thus xν = π2 and xν = π2 + 2νπ are maxinf-points of the Fν The sequence676
xννisinIN converges to a maxinf-point of F the second sequence xννisinIN does not677
678
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functions
References679
1 Attouch H Variational convergence for functions and operators Applicable Mathematics Series680
Pitman London (1984)681
2 Attouch H Wets RB Convergence des points minsup et de points fixes C R Acad Sci682
Paris 296 657ndash660 (1983)683
3 Attouch H Wets RB A convergence theory for saddle functions Trans Am Math Soc 280684
1ndash41 (1983)685
4 Aubin JP Ekeland I Applied nonlinear analysis Wiley London (1984)686
5 Aubin JP Frankowska H Set-valued analysis Birkhaumluser (1990)687
6 Bagh A Approximation for optimal control problems Lecture at Universidad de Chile Santiago688
(1999)689
7 Fan K A minimax inequality and applications In Shisha O (ed) InequalitiesmdashIII pp 103ndash113690
Academic Dublin (1972)691
8 Jofreacute A Rockafellar R Wets RB A variational inequality scheme for determining an economic692
equilibrium of classical or extended type In Giannessi F Maugeri A (eds) Variational analysis and693
applications pp 553ndash578 Springer New York (2005)694
9 Jofreacute A Wets RB Continuity properties of Walras equilibrium points Ann Oper Res 114 229ndash695
243 (2002)696
10 Jofreacute A Wets RB Variational convergence of bivariate functions Motivating applications II697
(Manuscript) (2006)698
11 Jofreacute A Wets RB Variational convergence of bivariate functions motivating applications I699
(Manuscript) (2004)700
12 Lignola M Morgan J Convergence of marginal functions with dependent con-701
straints Optimization 23 189ndash213 (1992)702
13 Rockafellar R Convex analysis Princeton University Press Princeton (1970)703
14 Rockafellar R Wets RB Variational analysis 2nd edn Springer New York (2004)704
123
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Metadata of the article that will be visualized in OnlineFirst
ArticleTitle Variational convergence of bivariate functions lopsided convergenceArticle Sub-Title
Journal Name Mathematical Programming
Corresponding Author Family Name WetsParticle
Given Name Roger J-BSuffix
Division Department of Mathematics
Organization University of California
Address Davis USA
Email rjbwetsucdavisedu
Author Family Name JofreacuteParticle
Given Name AlejandroSuffix
Division Ingeneria Matematica
Organization Universidad de Chile
Address Santiago Chile
Email ajofredimuchilecl
Schedule
Received 11 September 2005
Revised
Accepted 3 April 2006
Abstract We explore convergence notions for bivariate functions that yield convergence and stability results for theirmaxinf (or minsup) points This lays the foundations for the study of the stability of solutions to variationalinequalities the solutions of inclusions of Nash equilibrium points of non-cooperative games and Walraseconomic equilibrium points of fixed points of solutions to inclusions the primal and dual solutions ofconvex optimization problems and of zero-sum games These applications will be dealt with in a couple ofaccompanying papers
Keywords (separated by -) Lopsided convergence - Maxinf-points - Ky Fan functions - Variational inequalities - Epi-convergence
Mathematics SubjectClassification (2000)(separated by -)
65K10 - 90C31 - 91A10 - 47J20 - 47J30 - 49J45
Footnote Information Dedicated to A Auslender in recognition of his valuable contributions to Mathematical Programmingfoundations and numerical proceduresResearch supported in part by grants of the National Science Foundation and Fondap-Matematicas AplicadasUniversidad de Chile
Journal 10107 Article 122 Dear Author During the process of typesetting your article the following queries have arisen Please check your typeset proof carefully against the queries listed below and mark the necessary changes either directly on the proofonline grid or in the lsquoAuthorrsquos responsersquo area provided below
Author Query Form
Query Details required Authorrsquos response 1 Please check the author name
ldquoRoger J -B Wetsrdquo is correctly tagged (given and family name)
unco
rrec
ted
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f
Math Program Ser BDOI 101007s10107-007-0122-8
FULL LENGTH PAPER
Variational convergence of bivariate functions lopsidedconvergence
Alejandro Jofreacute middot Roger J-B Wets
Received 11 September 2005 Accepted 3 April 2006copy Springer-Verlag 2007
Abstract We explore convergence notions for bivariate functions that yield1
convergence and stability results for their maxinf (or minsup) points This lays the2
foundations for the study of the stability of solutions to variational inequalities the3
solutions of inclusions of Nash equilibrium points of non-cooperative games and4
Walras economic equilibrium points of fixed points of solutions to inclusions the5
primal and dual solutions of convex optimization problems and of zero-sum games6
These applications will be dealt with in a couple of accompanying papers7
Keywords Lopsided convergence middot Maxinf-points middot Ky Fan functions middot Variational8
points of non-cooperative games equilibrium points of zero-sum games etc We make33
a distinction between the situations when the bivariate function is generated from a34
single-valued mapping [11] or when the mapping can also be set-valued [10]35
The major tool is the notion of lopsided convergence that was introduced in [2]36
but is modified here so that a wider class of applications can be handled The major37
adjustment is that bivariate functions are not as in [2] no longer defined on all of38
Rn timesR
m with values in the extended reals but are now only finite-valued on a specific39
product C times D with C D subsets of Rn and R
m Dealing with lsquogeneralrsquo bivariate40
functions defined on the full product space was in keeping with the elegant work41
of Rockafellar [13] on duality relations for convexndashconcave bivariate functions and42
the subsequent work [3] on the epihypo-convergence of saddle functions However43
our present analysis actually shows that notwithstanding its esthetic allurement one44
should not cast bivariate functions even in the convexndashconcave case in the general45
extended-real valued framework In some way this is in contradiction with the uni-46
variate case where the extension by allowing for the values plusmninfin of functions defined47
on a (constrained) set to all of Rn has been so effectively exploited to derive a lsquounifiedrsquo48
convergence and differentiation theory [514] We shall show that some of this can be49
recovered but one must first make a clear distinction between max-inf problems and50
min-sup ones and only then one can generate the appropriate extensions after all51
also in the univariate case one makes a clear distinction when extending a function in52
a minimization setting or a maximization setting53
In order to be consistent in our presentation and to set up the results required54
later on we begin by a presentation of the theory of epi-convergence for real-valued55
univariate functions that are only defined on a subset of Rn No lsquonewrsquo results are56
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
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Variational convergence of bivariate functions
actually derived although a revised formulation is required We make the connection57
with the standard approach ie when these (univariate) functions are extended real-58
valued We then turn to lopsided convergence and point out the shortcomings of an59
lsquoextended real-valuedrsquo approach Finally we exploit our convergence result to obtain60
a extension of Ky Fan inequality [7] to situations when the domain of definition of the61
bivariate function is not necessarily compact62
2 Epi-convergence63
One can always represent an optimization problem involving constraints or not as64
one of minimizing an extended real-valued function In the case of a constrained-65
minimization problem simply redefine the objective as taking on the value infin out-66
side the feasible region the set determined by the constraints In this framework the67
canonical problem can be formulated as one of minimizing on all of Rn an extended68
real-valued function f Rn rarr R Approximation issues can consequently be stud-69
ied in terms of the convergence of such functions This has lead to the notion of70
epi-convergence1 that plays a key role in lsquoVariational Analysisrsquo [1514] when deal-71
ing with a maximization problem it is hypo-convergence the convergence of the72
hypographs that is the appropriate convergence notion73
Henceforth we restrict our development to the lsquominimization settingrsquo but at the74
end of this section we translate results and observations to the lsquomaximizationrsquo case75
As already indicated in Variational Analysis one usually deals with76
fcn(Rn) = f R
n rarr R
77
the space of extended real-valued functions that are defined on all of Rn even allowing78
for the possibility that they are nowhere finite-valued Definitions properties limits79
etc generally do not refer to the domain on which they are finite For reasons that will80
become clearer when we deal with the convergence of bivariate functions we need to81
depart from this simple and very convenient paradigm Our focus will be on82
f v-fcn(Rn) = f D rarr R
∣∣ for some empty = D sub R
n83
the class of all finite-valued functions with non-empty domain D sub Rn It must be84
understood that in this notation Rn does not refer to the domain of definition but to85
the underlying space that contains the domains on which the functions are defined86
The epigraph of a function f is always the set of all points in Rn+1 that lie on87
or above the graph of f irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) If88
f D rarr R belongs to f v-fcn(Rn) then89
epi f = (x α) isin D times R
∣∣ α ge f (x)
sub Rn+190
1 For extensive references and a survey of the field one can consult [15] and in particular the Commentarysection that concludes [14 Chap 7]
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
and if f belongs to fcn(Rn) then91
epi f = (x α) isin R
n+1∣∣α ge f (x)
92
A function f is lsc (=lower semicontinuous) if its epigraph is closed as a subset of93
Rn+1 ie epi f = cl(epi f ) with cl denoting closure [14 Theorem 16]294
So when f isin f v-fcn(Rn) lsc implies3 that for all xν isin D rarr x 95
ndash if x isin D liminfν f (xν) ge f (x) and96
ndash if x isin cl D D f (xν) rarr infin97
In our lsquominimizationrsquo framework cl f denotes the function whose epigraph is the98
closure relative to Rn+1 of the epigraph of f ie the lsc-regularization of f Its99
possible that when f isin f v-fcn cl f might be defined on a set thats strictly larger than100
D but always contained in cl D101
Lets now turn to convergence issues Recall that set-convergence in the Painleveacutendash102
Kuratowski sense [14 Sect 4B] is defined as follows Cν rarr C sub Rn if103
ndash (a-set) all cluster points of a sequence
xν isin CννisinIN belong to C 104
ndash (b-set) for each x isin C one can find a sequence xν isin Cν rarr x 105
When just condition (a-set) holds then C is then the outer limit of the sequence106 Cν
νisinIN and when its just (b-set) that holds then C is the inner limit [14 Chap 4107
Sect 2] Note that whenever C is the limit the outer- or the inner-limit its closed108
[14 Proposition 44] and that C = empty if and only if the sequence Cν eventually lsquoes-109
capesrsquo from any bounded set [14 Corollary 411] Moreover if the sequenceCν
νisinIN110
consists of convex sets its inner limit and its limit if it exists are also convex [14111
Proposition 415]112
Definition 1 (epi-convergence) A sequence of functions
f ν ν isin IN whose113
domains lie in Rn epi-converges to a function f when epi f ν rarr epi f as subsets114
of Rn+1 again irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) One then writes115
f ν rarre f 116
Figure 1 provides an example of two functions f and f ν that are close to each117
other in terms of the distance between their epigraphsmdashie the distance between the118
location of the two jumpsmdashbut are pretty far from each other pointwise or with respect119
to the infin-normmdashie the size of the jumps120
Let f ννisinIN be a sequence of functions with domains in Rn When121
ndash epi f is the outer limit of epi f ννisinIN one refers to f as the lower epi-limit of the122
functions f ν 123
ndash epi f is the inner limit of the epi f ν one refers to f as the upper epi-limit of the124
functions f ν 125
Of course f is the epi-limit of the sequence if its both the lower and upper epi-limit126
2 Throughout its implicitly assumed that Rn is equipped with its usual Euclidiean topology
3 Indeed if liminfν f (xν) lt infin then for some subsequence νk f (xνk ) rarr α isin R and because epi f isclosed it implies that (x α) isin epi f which would place x in the domain of f contradicting x isin D
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functions
x
f
f
Fig 1 f and f ν epigraphically close to each other
Proposition 1 (properties of epi-limits) Let f ννisinIN be a sequence of functions with127
domains in Rn Then the lower and upper epi-limits and the epi-limit if it exists are128
all lsc Moreover if the functions f ν are convex so is the upper epi-limit and the129
epi-limit if it exists130
Proof Follows immediately from the properties of set-limits 131
The last proposition implies in particular that the family of lsc functions is closed132
under epi-convergence133
The definition of epi-convergence for families of functions in fcn(Rn) is the usual134
one [14 Chap 7 Sect B] with all the implications concerning the convergence of135
the minimizers and infimal values [14 Chap 7 Sect E] But in a certain sense the136
definition is lsquonewrsquo when the focus is on epi-convergent families in f v-fcn(Rn) and137
its for this class of functions that we need to know the conditions under which one can138
claim convergence of the minimizers and infimums We chose to make the presentation139
self-contained although as will be shown later one could also embed f v-fcn(Rn) in140
a subclass of fcn(Rn) and then appeal to the lsquostandardrsquo results but unfortunately this141
requires that the non-initiated reader plows through a substantial amount of material142
When f is an epi-limit its necessarily a lsc function since its epigraph is the set-143
limit of a collection of sets in Rn+1 Its epigraph is closed but its domain D is not144
necessarily closed Simply think of the collection of functions f ν = f for all ν with145
D = (0infin) and f (x) = 1x on D This collection clearly epi-converges to the lsc146
function f on D with closed epigraph but not with closed domain147
Lemma 1 (epi-limit value at boundary points) Suppose f D rarr R is the epi-limit148
of a sequence
f ν Dν rarr RνisinIN with all functions in f v-fcn(Rn) Then for any149
sequence xν isin Dν rarr x liminfν f ν(xν) gt minusinfin150
Proof We proceed by contradiction Suppose that xν isin Dν rarr x and151
liminfν f ν(xν) = minusinfin By assumption f gt minusinfin on D thus the xν cannot converge152
to a point in D ie necessarily x isin D If thats the case and since epi f ν rarr epi f 153
the line x times R would have to lie in epi f contradicting the assumption that f the154
epi-limit of the f ν belongs to f v-fcn(Rn) 155
123
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Example 1 [an epi-limit thats not in f v-fcn(Rn)] Consider the sequence of functions156 f ν [0infin) rarr R
νisinIN with157
f ν(x) =
⎧⎪⎨
⎪⎩
minusν2x if 0 le x le νminus1
ν2x minus 2ν if νminus1 le x le 2νminus1
0 for x ge 2νminus1
158
Detail The functions f ν isin f v-fcn(R) and for the sequence xν = νminus1 f ν(xν) rarr minusinfin159
and f ν rarre f where f [0infin) rarr R with f equiv 0 on (0infin) and f (0) = minusinfin Thus the160
functions f ν epi-converge to f as functions in fcn(R) provided they are appropriately161
extended ie taking on the value infin on (minusinfin 0) But they do not epi-converge to a162
function in f v-fcn(R) 163
In addition to the lsquogeometricrsquo definition the next proposition provides an lsquoanalyticrsquo164
characterization of epi-converging sequences in f v-fcn(Rn)165
Proposition 2 [epi-convergence in f v-fcn(Rn)] Let
f D rarr R f ν Dν rarr166
R ν isin IN
be a collection of functions in f v-fcn(Rn) Then f ν rarre f if and only the167
following conditions are satisfied168
(a) forall xν isin Dν rarr x in D liminfν f ν(xν) ge f (x)169
(ainfin) for all xν isin Dν rarr x isin D f ν(xν)infin4170
(b) forallx isin D exist xν isin Dν rarr x such that limsupν f ν(xν) le f (x)171
Proof If epi f ν rarr epi f and xν isin Dν rarr x either lim infν f ν(xν) = α lt infin or172
not Lemma 1 reminds us that α = minusinfin is not a possibility In the first instance173
(x α) is a cluster point of(xν f ν(xν)) isin epi f ν
νisinIN and thus belongs to epi f 174
ie f (x) le α and hence (a) holds α gt minusinfin since otherwise f would not be finite175
valued on D If α = infin that means that f ν(xν)infin and x cannot belong to D176
and thus (ainfin) holds On the other hand if x isin D and thus f (x) is finite there is177
a(xν αν) isin epi f ν
νisinIN such that xν isin Dν rarr x isin D and αν rarr f (x) with178
αν ge f ν(xν) ie lim supν f ν(xν) le f (x) ie (b) is also satisfied179
Conversely if (a) and (ainfin) hold and (xν αν) isin epi f ν rarr (x α) then either x isin D180
or not recall also that in view of Lemma 1 α cannot be minusinfin since we are dealing181
with epi-convergence in f v-fcn(Rn) In the latter instance by (ainfin) α = infin so we are182
not dealing with a converging sequence of points (in Rn+1) and there is no need to183
consider this situation any further When x isin D since then lim infν f ν(xν) ge f (x)184
and αν ge f ν(xν) one has α ge f (x) and consequently (x α) belongs to epi f 185
this means that condition (a-set) is satisfied If (x α) isin epi f from (b) follows the186
existence of a sequence xν isin Dν rarr x such that lim supν f ν(xν) le f (x) le α We187
can then choose the αν ge f ν(xν) so that αν rarr α that yields (b-set) the second188
condition for the set-convergence of epi f ν rarr epi f 189
Theorem 1 (epi-convergence basic properties) Consider a sequence
f ν Dν rarr190
R ν isin IN sub f v-fcn(Rn) epi-converging to f D rarr R also in f v-fcn(Rn) Then191
4 means non-decreasing and converging to ie not necessarily monotonically
123
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Variational convergence of bivariate functions
lim supνrarrinfin
(inf f ν) le inf f192
Moreover if xk isin argminDνk f νk for some subsequence νk and xk rarr x then193
x isin argminD f and minDνk f νk rarr minD f 194
If argminD f is a singleton then every convergent subsequence of minimizers con-195
verges to argminD f 196
Proof Let xlinfinl=1 be a sequence in D such that f (xl) rarr inf f By 2(b) for each l197
one can find a sequence xνl isin Dν rarr xl such that lim supν f ν(xνl) le f (xl) Since198
for all ν inf f ν le f ν(xνl) it follows that for all l199
lim supν
(inf f ν) le lim supν
f ν(xνl) le f (xl)200
and one has lim supν(inf f ν) le inf f since f (xl) rarr inf f 201
For the sequence xk isin Dνk rarr x from the above and 2(a)202
inf f ge lim supk
f νk (xk) ge lim infk
f νk (xk) ge f (x)203
ie x minimizes f on D and f νk (xk) = minDνk f νk rarr minD f 204
Finally since every convergent subsequence of minimizers of the functions f ν205
converges to a minimizer of f it follows that it must converge to the unique minimizer206
when argminD f is a singleton 207
In most of the applications we shall rely on a somewhat more restrictive notion208
than lsquosimplersquo epi-convergence to guarantee the convergence of the infimums209
Definition 2 (tight epi-convergence) The sequence210
f ν Dν rarr RνisinIN sub f v-fcn(Rn) epi-converges tightly to f D rarr R isin211
f v-fcn(Rn) if f ν rarre f and for all ε gt 0 there exist a compact set Bε and an in-212
dex νε such that213
forall ν ge νε inf BεcapDν f ν le inf Dν f ν + ε214
Theorem 2 (convergence of the infimums) Let f ν Dν rarr RνisinIN sub f v-fcn(Rn)215
be a sequence of functions that epi-converges to the function f D rarr R also in216
f v-fcn(Rn) with inf D f finite Then they epi-converge tightly217
(a) if and only if inf Dν f ν rarr inf D f 218
(b) if and only if there exists a sequence εν 0 such that εν-argmin f ν rarr argmin f 219
Proof Lets start with necessity in (a) For given ε gt 0 the assumptions and Theorem 1220
imply221
lim infν
( infDνcapBε
f ν) le lim infν
(infDν
f ν) + ε le lim supν
(infDν
f ν) + ε le infD
f + ε lt infin222
123
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If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223
follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224
point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225
with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226
it follows that there is no such sequences with κ arbitrarily negative In other words227
excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228
from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229
a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230
Hence231
inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232
In combination with our first string of inequalities and the fact that ε gt 0 can be233
chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234
Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235
enough given any δ gt 0 to exhibit a compact set B such that lim supν
(inf BcapDν f ν
) le236
inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237
f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238
Let B be any compact set large enough to contain all the points xν Then inf B f ν le239
f ν(xν) for all ν so B has the desired property240
We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241
tion and consequently for ν large enough also αν is finite Since convergence of242
the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243
find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244
εν = αν minus αν 245
For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246
f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247
because f ν rarre f one obtains248
inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249
le lim infν(inf f ν) le limsupν(inf f ν) le inf f250
where the last inequality comes from Theorem 1 251
Remark 1 (convergence of domains) Although epi-convergence essentially implies252
convergence of the level sets [14 Proposition 77] it does not follow that it implies253
the convergence of their (effective) domains Indeed consider the following sequence254
f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255
function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256
This vigorously argues against the temptation of involving the convergence of their257
domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258
This concludes the presentation of the results that will be used in the sequel As259
indicated earlier its also possible to derive these results from those for extended real-260
valued functions To do so one identifies f v-fcn(Rn) with261
pr -fcn(Rn) = f isin fcn(Rn)
∣∣ minus infin lt f equiv infin
262
123
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Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
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η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
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Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
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Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
the functions gν hypo-converge to g Maxinf-points for Fν and F are then maximizers599
of the corresponding functions gν and g The assertions now follow immediately600
from the convergence of the argmax of hypo-converging sequences cf Theorem [14601
Theorem 731] translated to the lsquomaximizationrsquo framework 602
To deal with a lsquominsuprsquo situations one can either repeat all the arguments changing603
inf to sup liminf to limsup and vice-versa or simply re-integrate the questions to604
the lsquomaxinfrsquo framework by changing signs of the approximating and limit bivariate605
functions606
123
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Variational convergence of bivariate functions
4 Ky Fanrsquos Inequality extended607
The class of usc functions is closed under hypo-converge [14 Theorem 74] and so608
is the class of concave usc functions [14 Theorem 717] A class of functions that609
is closed under lopsided convergence is the class of Ky Fan functions (Theorem 8)610
We exploit this result to obtain a generalization of Ky Fan Inequality that allows us611
to claim existence of maxinf-points in situations when the domain of definition of the612
Ky Fan function is not necessarily compact613
Definition 7 A bivariate function F C times D rarr R with Cand D convex sets in614
f v-biv(Rn+m) is called a Ky Fan function if615
(a) forall y isin D x rarr F(x y) is usc on C 616
(b) forall x isin C y rarr F(x y) is convex on D617
Note that the sets C or D are not required to be compact618
Theorem 8 (lop-limits of Ky Fan functions) The lopsided limit F C times D rarr R of a619
sequence
Fν Cν times Dν rarr RνisinIN of Ky Fan functions in f v-biv(Rn+m) is also a620
Ky Fan function621
Proof For the convexity of y rarr F(x y) let xν isin Cν rarr x isin C be the sequence set622
forth by 3(b) and y0 yλ y1 isin D with yλ = (1 minus λ)y0 + λy1 for λ isin [0 1] In view623
of 3(a) we can choose sequences
y0ν isin Dν rarr y0
y1ν isin Dν rarr y1
such that624
Fν(xν y0ν) rarr F(x y0) and Fν(xν y1ν) rarr F(x y1) Let yλν = (1 minus λ)y0ν +625
λy1ν yλν isin Dν since the functions Fν(x middot) are convex and the sequence yλννisinIN626
certainly converges to yλ For all ν one has627
Fν(xν yλν) le (1 minus λ)Fν(xν y0ν) + λFν(xν y1ν)628
Taking lininf on both sides yields629
F(x yλ) le liminfν Fν(xν yλν) le (1 minus λ)F(x y0) + λF(x y1)630
that establishes the convexity of F(x middot)631
To prove the upper semicontinuity of F with respect to x-variable we show that632
for y isin D633
hypo F(middot y) is the inner set-limit of the hypo Fν(middot yν)634
where the limit is with respect to all sequences yν isin DννisinIN converging to y and635
ν rarr infin This yields the upper semicontinuity since the inner set-limit is always636
closed and a function is usc if and only if its hypograph is closed We have to show637
that if (x α) isin hypo F(middot y) then whenever yν isin Dν rarr y one can find (xν αν) isin638
hypo Fν(middot yν) such that (xν αν) rarr (x α) But that follows immediately from 3(a)639
since we can adjust the αν le Fν(xν yν) so that they converge to α le F(x y) 640
Given a Ky Fan function with compact domain and non-negative on the diagonal641
we have the following important existence result642
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
Lemma 2 (Ky Fanrsquos Inequality [7] [4 Theorem 635]) Suppose F C times C rarr R is643
a Ky Fan function with C compact Then the set of maxinf-points of F is a nonempty644
subset of C Moreover if F(x x) ge 0 (on C times C) then for every maxinf-point x of645
C F(x middot) ge 0 on C646
One of the consequences of the lopsided convergence is an extension of the Ky Fanrsquos647
Inequality to the case when it is not possible to apply it directly because one of the648
conditions is not satisfied for example the compactness of the domain However we are649
able to approach the bivariate function F by a sequence FννisinIN defined on compact650
sets Cν This procedure could be useful in many situation where the original maxinf-651
problem is unbounded and then the problem is approached by a family of truncated652
maxinf-problems Such is the case for example when we consider as variables in653
the original problem the multipliers associated to inequality constraints or when the654
original problem is a Walras equilibrium with a positive orthant as consumption set655
in [8] one is precisely confronted with such situations Another simple illustrative656
example follows the statement of the theorem657
Theorem 9 (Extension of Ky Fanrsquos Inequality) Let empty = C sub Rn and F a finite-valued658
bivariate function defined on CtimesC Suppose one can find sequences of compact convex659
setsCν sub R
n
and (finite-valued) Ky Fan functions Fν Cν times Cν rarr RνisinIN lop-660
converging y-tightly to F then every cluster point x of any sequence xν ν isin IN of661
maxinf-points of the Fν is a maxinf-point of F662
Proof Ky Fanrsquos Inequality 2 implies that for all ν the set of maxinf-points of Fν is663
non-empty On the other hand in view of Theorems 8 and 4 any cluster point of such664
maxinf-points will be a maxinf-point of F 665
Example 2 (Extended Ky Fanrsquos Inequality applied) We consider a Ky Fan function666
F(x y) = sin x + (y + 1)minus1 defined on the set [0infin)2 Although667
inf yisin[0infin) F(x y) = sin x668
and the set maxinf-points is not empty we cannot apply Ky Fan Inequality because669
the domain of F is not compact the function F(middot y) is not even sup-compact670
Detail If we consider the functions Fν(x y) = sin x + (y + 1)minus1 on the compact671
domains [0 ν]2 one can apply Ky Fanrsquos Inequality Indeed in this case we have672
inf yisin[0ν) F(x y) = sin x + (ν + 1)minus1673
that converges pointwise and hypo- to sin x and674
argmaxyisin[0ν] inf yisin[0ν] F(x y) = π2 + 2kπ
∣∣ k isin IN
675
Thus xν = π2 and xν = π2 + 2νπ are maxinf-points of the Fν The sequence676
xννisinIN converges to a maxinf-point of F the second sequence xννisinIN does not677
678
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functions
References679
1 Attouch H Variational convergence for functions and operators Applicable Mathematics Series680
Pitman London (1984)681
2 Attouch H Wets RB Convergence des points minsup et de points fixes C R Acad Sci682
Paris 296 657ndash660 (1983)683
3 Attouch H Wets RB A convergence theory for saddle functions Trans Am Math Soc 280684
1ndash41 (1983)685
4 Aubin JP Ekeland I Applied nonlinear analysis Wiley London (1984)686
5 Aubin JP Frankowska H Set-valued analysis Birkhaumluser (1990)687
6 Bagh A Approximation for optimal control problems Lecture at Universidad de Chile Santiago688
(1999)689
7 Fan K A minimax inequality and applications In Shisha O (ed) InequalitiesmdashIII pp 103ndash113690
Academic Dublin (1972)691
8 Jofreacute A Rockafellar R Wets RB A variational inequality scheme for determining an economic692
equilibrium of classical or extended type In Giannessi F Maugeri A (eds) Variational analysis and693
applications pp 553ndash578 Springer New York (2005)694
9 Jofreacute A Wets RB Continuity properties of Walras equilibrium points Ann Oper Res 114 229ndash695
243 (2002)696
10 Jofreacute A Wets RB Variational convergence of bivariate functions Motivating applications II697
(Manuscript) (2006)698
11 Jofreacute A Wets RB Variational convergence of bivariate functions motivating applications I699
(Manuscript) (2004)700
12 Lignola M Morgan J Convergence of marginal functions with dependent con-701
straints Optimization 23 189ndash213 (1992)702
13 Rockafellar R Convex analysis Princeton University Press Princeton (1970)703
14 Rockafellar R Wets RB Variational analysis 2nd edn Springer New York (2004)704
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
Journal 10107 Article 122 Dear Author During the process of typesetting your article the following queries have arisen Please check your typeset proof carefully against the queries listed below and mark the necessary changes either directly on the proofonline grid or in the lsquoAuthorrsquos responsersquo area provided below
Author Query Form
Query Details required Authorrsquos response 1 Please check the author name
ldquoRoger J -B Wetsrdquo is correctly tagged (given and family name)
unco
rrec
ted
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Math Program Ser BDOI 101007s10107-007-0122-8
FULL LENGTH PAPER
Variational convergence of bivariate functions lopsidedconvergence
Alejandro Jofreacute middot Roger J-B Wets
Received 11 September 2005 Accepted 3 April 2006copy Springer-Verlag 2007
Abstract We explore convergence notions for bivariate functions that yield1
convergence and stability results for their maxinf (or minsup) points This lays the2
foundations for the study of the stability of solutions to variational inequalities the3
solutions of inclusions of Nash equilibrium points of non-cooperative games and4
Walras economic equilibrium points of fixed points of solutions to inclusions the5
primal and dual solutions of convex optimization problems and of zero-sum games6
These applications will be dealt with in a couple of accompanying papers7
Keywords Lopsided convergence middot Maxinf-points middot Ky Fan functions middot Variational8
points of non-cooperative games equilibrium points of zero-sum games etc We make33
a distinction between the situations when the bivariate function is generated from a34
single-valued mapping [11] or when the mapping can also be set-valued [10]35
The major tool is the notion of lopsided convergence that was introduced in [2]36
but is modified here so that a wider class of applications can be handled The major37
adjustment is that bivariate functions are not as in [2] no longer defined on all of38
Rn timesR
m with values in the extended reals but are now only finite-valued on a specific39
product C times D with C D subsets of Rn and R
m Dealing with lsquogeneralrsquo bivariate40
functions defined on the full product space was in keeping with the elegant work41
of Rockafellar [13] on duality relations for convexndashconcave bivariate functions and42
the subsequent work [3] on the epihypo-convergence of saddle functions However43
our present analysis actually shows that notwithstanding its esthetic allurement one44
should not cast bivariate functions even in the convexndashconcave case in the general45
extended-real valued framework In some way this is in contradiction with the uni-46
variate case where the extension by allowing for the values plusmninfin of functions defined47
on a (constrained) set to all of Rn has been so effectively exploited to derive a lsquounifiedrsquo48
convergence and differentiation theory [514] We shall show that some of this can be49
recovered but one must first make a clear distinction between max-inf problems and50
min-sup ones and only then one can generate the appropriate extensions after all51
also in the univariate case one makes a clear distinction when extending a function in52
a minimization setting or a maximization setting53
In order to be consistent in our presentation and to set up the results required54
later on we begin by a presentation of the theory of epi-convergence for real-valued55
univariate functions that are only defined on a subset of Rn No lsquonewrsquo results are56
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functions
actually derived although a revised formulation is required We make the connection57
with the standard approach ie when these (univariate) functions are extended real-58
valued We then turn to lopsided convergence and point out the shortcomings of an59
lsquoextended real-valuedrsquo approach Finally we exploit our convergence result to obtain60
a extension of Ky Fan inequality [7] to situations when the domain of definition of the61
bivariate function is not necessarily compact62
2 Epi-convergence63
One can always represent an optimization problem involving constraints or not as64
one of minimizing an extended real-valued function In the case of a constrained-65
minimization problem simply redefine the objective as taking on the value infin out-66
side the feasible region the set determined by the constraints In this framework the67
canonical problem can be formulated as one of minimizing on all of Rn an extended68
real-valued function f Rn rarr R Approximation issues can consequently be stud-69
ied in terms of the convergence of such functions This has lead to the notion of70
epi-convergence1 that plays a key role in lsquoVariational Analysisrsquo [1514] when deal-71
ing with a maximization problem it is hypo-convergence the convergence of the72
hypographs that is the appropriate convergence notion73
Henceforth we restrict our development to the lsquominimization settingrsquo but at the74
end of this section we translate results and observations to the lsquomaximizationrsquo case75
As already indicated in Variational Analysis one usually deals with76
fcn(Rn) = f R
n rarr R
77
the space of extended real-valued functions that are defined on all of Rn even allowing78
for the possibility that they are nowhere finite-valued Definitions properties limits79
etc generally do not refer to the domain on which they are finite For reasons that will80
become clearer when we deal with the convergence of bivariate functions we need to81
depart from this simple and very convenient paradigm Our focus will be on82
f v-fcn(Rn) = f D rarr R
∣∣ for some empty = D sub R
n83
the class of all finite-valued functions with non-empty domain D sub Rn It must be84
understood that in this notation Rn does not refer to the domain of definition but to85
the underlying space that contains the domains on which the functions are defined86
The epigraph of a function f is always the set of all points in Rn+1 that lie on87
or above the graph of f irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) If88
f D rarr R belongs to f v-fcn(Rn) then89
epi f = (x α) isin D times R
∣∣ α ge f (x)
sub Rn+190
1 For extensive references and a survey of the field one can consult [15] and in particular the Commentarysection that concludes [14 Chap 7]
123
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A Jofreacute R J -B Wets
and if f belongs to fcn(Rn) then91
epi f = (x α) isin R
n+1∣∣α ge f (x)
92
A function f is lsc (=lower semicontinuous) if its epigraph is closed as a subset of93
Rn+1 ie epi f = cl(epi f ) with cl denoting closure [14 Theorem 16]294
So when f isin f v-fcn(Rn) lsc implies3 that for all xν isin D rarr x 95
ndash if x isin D liminfν f (xν) ge f (x) and96
ndash if x isin cl D D f (xν) rarr infin97
In our lsquominimizationrsquo framework cl f denotes the function whose epigraph is the98
closure relative to Rn+1 of the epigraph of f ie the lsc-regularization of f Its99
possible that when f isin f v-fcn cl f might be defined on a set thats strictly larger than100
D but always contained in cl D101
Lets now turn to convergence issues Recall that set-convergence in the Painleveacutendash102
Kuratowski sense [14 Sect 4B] is defined as follows Cν rarr C sub Rn if103
ndash (a-set) all cluster points of a sequence
xν isin CννisinIN belong to C 104
ndash (b-set) for each x isin C one can find a sequence xν isin Cν rarr x 105
When just condition (a-set) holds then C is then the outer limit of the sequence106 Cν
νisinIN and when its just (b-set) that holds then C is the inner limit [14 Chap 4107
Sect 2] Note that whenever C is the limit the outer- or the inner-limit its closed108
[14 Proposition 44] and that C = empty if and only if the sequence Cν eventually lsquoes-109
capesrsquo from any bounded set [14 Corollary 411] Moreover if the sequenceCν
νisinIN110
consists of convex sets its inner limit and its limit if it exists are also convex [14111
Proposition 415]112
Definition 1 (epi-convergence) A sequence of functions
f ν ν isin IN whose113
domains lie in Rn epi-converges to a function f when epi f ν rarr epi f as subsets114
of Rn+1 again irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) One then writes115
f ν rarre f 116
Figure 1 provides an example of two functions f and f ν that are close to each117
other in terms of the distance between their epigraphsmdashie the distance between the118
location of the two jumpsmdashbut are pretty far from each other pointwise or with respect119
to the infin-normmdashie the size of the jumps120
Let f ννisinIN be a sequence of functions with domains in Rn When121
ndash epi f is the outer limit of epi f ννisinIN one refers to f as the lower epi-limit of the122
functions f ν 123
ndash epi f is the inner limit of the epi f ν one refers to f as the upper epi-limit of the124
functions f ν 125
Of course f is the epi-limit of the sequence if its both the lower and upper epi-limit126
2 Throughout its implicitly assumed that Rn is equipped with its usual Euclidiean topology
3 Indeed if liminfν f (xν) lt infin then for some subsequence νk f (xνk ) rarr α isin R and because epi f isclosed it implies that (x α) isin epi f which would place x in the domain of f contradicting x isin D
123
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Variational convergence of bivariate functions
x
f
f
Fig 1 f and f ν epigraphically close to each other
Proposition 1 (properties of epi-limits) Let f ννisinIN be a sequence of functions with127
domains in Rn Then the lower and upper epi-limits and the epi-limit if it exists are128
all lsc Moreover if the functions f ν are convex so is the upper epi-limit and the129
epi-limit if it exists130
Proof Follows immediately from the properties of set-limits 131
The last proposition implies in particular that the family of lsc functions is closed132
under epi-convergence133
The definition of epi-convergence for families of functions in fcn(Rn) is the usual134
one [14 Chap 7 Sect B] with all the implications concerning the convergence of135
the minimizers and infimal values [14 Chap 7 Sect E] But in a certain sense the136
definition is lsquonewrsquo when the focus is on epi-convergent families in f v-fcn(Rn) and137
its for this class of functions that we need to know the conditions under which one can138
claim convergence of the minimizers and infimums We chose to make the presentation139
self-contained although as will be shown later one could also embed f v-fcn(Rn) in140
a subclass of fcn(Rn) and then appeal to the lsquostandardrsquo results but unfortunately this141
requires that the non-initiated reader plows through a substantial amount of material142
When f is an epi-limit its necessarily a lsc function since its epigraph is the set-143
limit of a collection of sets in Rn+1 Its epigraph is closed but its domain D is not144
necessarily closed Simply think of the collection of functions f ν = f for all ν with145
D = (0infin) and f (x) = 1x on D This collection clearly epi-converges to the lsc146
function f on D with closed epigraph but not with closed domain147
Lemma 1 (epi-limit value at boundary points) Suppose f D rarr R is the epi-limit148
of a sequence
f ν Dν rarr RνisinIN with all functions in f v-fcn(Rn) Then for any149
sequence xν isin Dν rarr x liminfν f ν(xν) gt minusinfin150
Proof We proceed by contradiction Suppose that xν isin Dν rarr x and151
liminfν f ν(xν) = minusinfin By assumption f gt minusinfin on D thus the xν cannot converge152
to a point in D ie necessarily x isin D If thats the case and since epi f ν rarr epi f 153
the line x times R would have to lie in epi f contradicting the assumption that f the154
epi-limit of the f ν belongs to f v-fcn(Rn) 155
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
Example 1 [an epi-limit thats not in f v-fcn(Rn)] Consider the sequence of functions156 f ν [0infin) rarr R
νisinIN with157
f ν(x) =
⎧⎪⎨
⎪⎩
minusν2x if 0 le x le νminus1
ν2x minus 2ν if νminus1 le x le 2νminus1
0 for x ge 2νminus1
158
Detail The functions f ν isin f v-fcn(R) and for the sequence xν = νminus1 f ν(xν) rarr minusinfin159
and f ν rarre f where f [0infin) rarr R with f equiv 0 on (0infin) and f (0) = minusinfin Thus the160
functions f ν epi-converge to f as functions in fcn(R) provided they are appropriately161
extended ie taking on the value infin on (minusinfin 0) But they do not epi-converge to a162
function in f v-fcn(R) 163
In addition to the lsquogeometricrsquo definition the next proposition provides an lsquoanalyticrsquo164
characterization of epi-converging sequences in f v-fcn(Rn)165
Proposition 2 [epi-convergence in f v-fcn(Rn)] Let
f D rarr R f ν Dν rarr166
R ν isin IN
be a collection of functions in f v-fcn(Rn) Then f ν rarre f if and only the167
following conditions are satisfied168
(a) forall xν isin Dν rarr x in D liminfν f ν(xν) ge f (x)169
(ainfin) for all xν isin Dν rarr x isin D f ν(xν)infin4170
(b) forallx isin D exist xν isin Dν rarr x such that limsupν f ν(xν) le f (x)171
Proof If epi f ν rarr epi f and xν isin Dν rarr x either lim infν f ν(xν) = α lt infin or172
not Lemma 1 reminds us that α = minusinfin is not a possibility In the first instance173
(x α) is a cluster point of(xν f ν(xν)) isin epi f ν
νisinIN and thus belongs to epi f 174
ie f (x) le α and hence (a) holds α gt minusinfin since otherwise f would not be finite175
valued on D If α = infin that means that f ν(xν)infin and x cannot belong to D176
and thus (ainfin) holds On the other hand if x isin D and thus f (x) is finite there is177
a(xν αν) isin epi f ν
νisinIN such that xν isin Dν rarr x isin D and αν rarr f (x) with178
αν ge f ν(xν) ie lim supν f ν(xν) le f (x) ie (b) is also satisfied179
Conversely if (a) and (ainfin) hold and (xν αν) isin epi f ν rarr (x α) then either x isin D180
or not recall also that in view of Lemma 1 α cannot be minusinfin since we are dealing181
with epi-convergence in f v-fcn(Rn) In the latter instance by (ainfin) α = infin so we are182
not dealing with a converging sequence of points (in Rn+1) and there is no need to183
consider this situation any further When x isin D since then lim infν f ν(xν) ge f (x)184
and αν ge f ν(xν) one has α ge f (x) and consequently (x α) belongs to epi f 185
this means that condition (a-set) is satisfied If (x α) isin epi f from (b) follows the186
existence of a sequence xν isin Dν rarr x such that lim supν f ν(xν) le f (x) le α We187
can then choose the αν ge f ν(xν) so that αν rarr α that yields (b-set) the second188
condition for the set-convergence of epi f ν rarr epi f 189
Theorem 1 (epi-convergence basic properties) Consider a sequence
f ν Dν rarr190
R ν isin IN sub f v-fcn(Rn) epi-converging to f D rarr R also in f v-fcn(Rn) Then191
4 means non-decreasing and converging to ie not necessarily monotonically
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functions
lim supνrarrinfin
(inf f ν) le inf f192
Moreover if xk isin argminDνk f νk for some subsequence νk and xk rarr x then193
x isin argminD f and minDνk f νk rarr minD f 194
If argminD f is a singleton then every convergent subsequence of minimizers con-195
verges to argminD f 196
Proof Let xlinfinl=1 be a sequence in D such that f (xl) rarr inf f By 2(b) for each l197
one can find a sequence xνl isin Dν rarr xl such that lim supν f ν(xνl) le f (xl) Since198
for all ν inf f ν le f ν(xνl) it follows that for all l199
lim supν
(inf f ν) le lim supν
f ν(xνl) le f (xl)200
and one has lim supν(inf f ν) le inf f since f (xl) rarr inf f 201
For the sequence xk isin Dνk rarr x from the above and 2(a)202
inf f ge lim supk
f νk (xk) ge lim infk
f νk (xk) ge f (x)203
ie x minimizes f on D and f νk (xk) = minDνk f νk rarr minD f 204
Finally since every convergent subsequence of minimizers of the functions f ν205
converges to a minimizer of f it follows that it must converge to the unique minimizer206
when argminD f is a singleton 207
In most of the applications we shall rely on a somewhat more restrictive notion208
than lsquosimplersquo epi-convergence to guarantee the convergence of the infimums209
Definition 2 (tight epi-convergence) The sequence210
f ν Dν rarr RνisinIN sub f v-fcn(Rn) epi-converges tightly to f D rarr R isin211
f v-fcn(Rn) if f ν rarre f and for all ε gt 0 there exist a compact set Bε and an in-212
dex νε such that213
forall ν ge νε inf BεcapDν f ν le inf Dν f ν + ε214
Theorem 2 (convergence of the infimums) Let f ν Dν rarr RνisinIN sub f v-fcn(Rn)215
be a sequence of functions that epi-converges to the function f D rarr R also in216
f v-fcn(Rn) with inf D f finite Then they epi-converge tightly217
(a) if and only if inf Dν f ν rarr inf D f 218
(b) if and only if there exists a sequence εν 0 such that εν-argmin f ν rarr argmin f 219
Proof Lets start with necessity in (a) For given ε gt 0 the assumptions and Theorem 1220
imply221
lim infν
( infDνcapBε
f ν) le lim infν
(infDν
f ν) + ε le lim supν
(infDν
f ν) + ε le infD
f + ε lt infin222
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223
follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224
point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225
with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226
it follows that there is no such sequences with κ arbitrarily negative In other words227
excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228
from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229
a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230
Hence231
inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232
In combination with our first string of inequalities and the fact that ε gt 0 can be233
chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234
Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235
enough given any δ gt 0 to exhibit a compact set B such that lim supν
(inf BcapDν f ν
) le236
inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237
f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238
Let B be any compact set large enough to contain all the points xν Then inf B f ν le239
f ν(xν) for all ν so B has the desired property240
We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241
tion and consequently for ν large enough also αν is finite Since convergence of242
the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243
find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244
εν = αν minus αν 245
For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246
f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247
because f ν rarre f one obtains248
inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249
le lim infν(inf f ν) le limsupν(inf f ν) le inf f250
where the last inequality comes from Theorem 1 251
Remark 1 (convergence of domains) Although epi-convergence essentially implies252
convergence of the level sets [14 Proposition 77] it does not follow that it implies253
the convergence of their (effective) domains Indeed consider the following sequence254
f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255
function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256
This vigorously argues against the temptation of involving the convergence of their257
domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258
This concludes the presentation of the results that will be used in the sequel As259
indicated earlier its also possible to derive these results from those for extended real-260
valued functions To do so one identifies f v-fcn(Rn) with261
pr -fcn(Rn) = f isin fcn(Rn)
∣∣ minus infin lt f equiv infin
262
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
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f
Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
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ted
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A Jofreacute R J -B Wets
η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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ted
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f
Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
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Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
points of non-cooperative games equilibrium points of zero-sum games etc We make33
a distinction between the situations when the bivariate function is generated from a34
single-valued mapping [11] or when the mapping can also be set-valued [10]35
The major tool is the notion of lopsided convergence that was introduced in [2]36
but is modified here so that a wider class of applications can be handled The major37
adjustment is that bivariate functions are not as in [2] no longer defined on all of38
Rn timesR
m with values in the extended reals but are now only finite-valued on a specific39
product C times D with C D subsets of Rn and R
m Dealing with lsquogeneralrsquo bivariate40
functions defined on the full product space was in keeping with the elegant work41
of Rockafellar [13] on duality relations for convexndashconcave bivariate functions and42
the subsequent work [3] on the epihypo-convergence of saddle functions However43
our present analysis actually shows that notwithstanding its esthetic allurement one44
should not cast bivariate functions even in the convexndashconcave case in the general45
extended-real valued framework In some way this is in contradiction with the uni-46
variate case where the extension by allowing for the values plusmninfin of functions defined47
on a (constrained) set to all of Rn has been so effectively exploited to derive a lsquounifiedrsquo48
convergence and differentiation theory [514] We shall show that some of this can be49
recovered but one must first make a clear distinction between max-inf problems and50
min-sup ones and only then one can generate the appropriate extensions after all51
also in the univariate case one makes a clear distinction when extending a function in52
a minimization setting or a maximization setting53
In order to be consistent in our presentation and to set up the results required54
later on we begin by a presentation of the theory of epi-convergence for real-valued55
univariate functions that are only defined on a subset of Rn No lsquonewrsquo results are56
123
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Variational convergence of bivariate functions
actually derived although a revised formulation is required We make the connection57
with the standard approach ie when these (univariate) functions are extended real-58
valued We then turn to lopsided convergence and point out the shortcomings of an59
lsquoextended real-valuedrsquo approach Finally we exploit our convergence result to obtain60
a extension of Ky Fan inequality [7] to situations when the domain of definition of the61
bivariate function is not necessarily compact62
2 Epi-convergence63
One can always represent an optimization problem involving constraints or not as64
one of minimizing an extended real-valued function In the case of a constrained-65
minimization problem simply redefine the objective as taking on the value infin out-66
side the feasible region the set determined by the constraints In this framework the67
canonical problem can be formulated as one of minimizing on all of Rn an extended68
real-valued function f Rn rarr R Approximation issues can consequently be stud-69
ied in terms of the convergence of such functions This has lead to the notion of70
epi-convergence1 that plays a key role in lsquoVariational Analysisrsquo [1514] when deal-71
ing with a maximization problem it is hypo-convergence the convergence of the72
hypographs that is the appropriate convergence notion73
Henceforth we restrict our development to the lsquominimization settingrsquo but at the74
end of this section we translate results and observations to the lsquomaximizationrsquo case75
As already indicated in Variational Analysis one usually deals with76
fcn(Rn) = f R
n rarr R
77
the space of extended real-valued functions that are defined on all of Rn even allowing78
for the possibility that they are nowhere finite-valued Definitions properties limits79
etc generally do not refer to the domain on which they are finite For reasons that will80
become clearer when we deal with the convergence of bivariate functions we need to81
depart from this simple and very convenient paradigm Our focus will be on82
f v-fcn(Rn) = f D rarr R
∣∣ for some empty = D sub R
n83
the class of all finite-valued functions with non-empty domain D sub Rn It must be84
understood that in this notation Rn does not refer to the domain of definition but to85
the underlying space that contains the domains on which the functions are defined86
The epigraph of a function f is always the set of all points in Rn+1 that lie on87
or above the graph of f irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) If88
f D rarr R belongs to f v-fcn(Rn) then89
epi f = (x α) isin D times R
∣∣ α ge f (x)
sub Rn+190
1 For extensive references and a survey of the field one can consult [15] and in particular the Commentarysection that concludes [14 Chap 7]
123
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f
A Jofreacute R J -B Wets
and if f belongs to fcn(Rn) then91
epi f = (x α) isin R
n+1∣∣α ge f (x)
92
A function f is lsc (=lower semicontinuous) if its epigraph is closed as a subset of93
Rn+1 ie epi f = cl(epi f ) with cl denoting closure [14 Theorem 16]294
So when f isin f v-fcn(Rn) lsc implies3 that for all xν isin D rarr x 95
ndash if x isin D liminfν f (xν) ge f (x) and96
ndash if x isin cl D D f (xν) rarr infin97
In our lsquominimizationrsquo framework cl f denotes the function whose epigraph is the98
closure relative to Rn+1 of the epigraph of f ie the lsc-regularization of f Its99
possible that when f isin f v-fcn cl f might be defined on a set thats strictly larger than100
D but always contained in cl D101
Lets now turn to convergence issues Recall that set-convergence in the Painleveacutendash102
Kuratowski sense [14 Sect 4B] is defined as follows Cν rarr C sub Rn if103
ndash (a-set) all cluster points of a sequence
xν isin CννisinIN belong to C 104
ndash (b-set) for each x isin C one can find a sequence xν isin Cν rarr x 105
When just condition (a-set) holds then C is then the outer limit of the sequence106 Cν
νisinIN and when its just (b-set) that holds then C is the inner limit [14 Chap 4107
Sect 2] Note that whenever C is the limit the outer- or the inner-limit its closed108
[14 Proposition 44] and that C = empty if and only if the sequence Cν eventually lsquoes-109
capesrsquo from any bounded set [14 Corollary 411] Moreover if the sequenceCν
νisinIN110
consists of convex sets its inner limit and its limit if it exists are also convex [14111
Proposition 415]112
Definition 1 (epi-convergence) A sequence of functions
f ν ν isin IN whose113
domains lie in Rn epi-converges to a function f when epi f ν rarr epi f as subsets114
of Rn+1 again irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) One then writes115
f ν rarre f 116
Figure 1 provides an example of two functions f and f ν that are close to each117
other in terms of the distance between their epigraphsmdashie the distance between the118
location of the two jumpsmdashbut are pretty far from each other pointwise or with respect119
to the infin-normmdashie the size of the jumps120
Let f ννisinIN be a sequence of functions with domains in Rn When121
ndash epi f is the outer limit of epi f ννisinIN one refers to f as the lower epi-limit of the122
functions f ν 123
ndash epi f is the inner limit of the epi f ν one refers to f as the upper epi-limit of the124
functions f ν 125
Of course f is the epi-limit of the sequence if its both the lower and upper epi-limit126
2 Throughout its implicitly assumed that Rn is equipped with its usual Euclidiean topology
3 Indeed if liminfν f (xν) lt infin then for some subsequence νk f (xνk ) rarr α isin R and because epi f isclosed it implies that (x α) isin epi f which would place x in the domain of f contradicting x isin D
123
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Variational convergence of bivariate functions
x
f
f
Fig 1 f and f ν epigraphically close to each other
Proposition 1 (properties of epi-limits) Let f ννisinIN be a sequence of functions with127
domains in Rn Then the lower and upper epi-limits and the epi-limit if it exists are128
all lsc Moreover if the functions f ν are convex so is the upper epi-limit and the129
epi-limit if it exists130
Proof Follows immediately from the properties of set-limits 131
The last proposition implies in particular that the family of lsc functions is closed132
under epi-convergence133
The definition of epi-convergence for families of functions in fcn(Rn) is the usual134
one [14 Chap 7 Sect B] with all the implications concerning the convergence of135
the minimizers and infimal values [14 Chap 7 Sect E] But in a certain sense the136
definition is lsquonewrsquo when the focus is on epi-convergent families in f v-fcn(Rn) and137
its for this class of functions that we need to know the conditions under which one can138
claim convergence of the minimizers and infimums We chose to make the presentation139
self-contained although as will be shown later one could also embed f v-fcn(Rn) in140
a subclass of fcn(Rn) and then appeal to the lsquostandardrsquo results but unfortunately this141
requires that the non-initiated reader plows through a substantial amount of material142
When f is an epi-limit its necessarily a lsc function since its epigraph is the set-143
limit of a collection of sets in Rn+1 Its epigraph is closed but its domain D is not144
necessarily closed Simply think of the collection of functions f ν = f for all ν with145
D = (0infin) and f (x) = 1x on D This collection clearly epi-converges to the lsc146
function f on D with closed epigraph but not with closed domain147
Lemma 1 (epi-limit value at boundary points) Suppose f D rarr R is the epi-limit148
of a sequence
f ν Dν rarr RνisinIN with all functions in f v-fcn(Rn) Then for any149
sequence xν isin Dν rarr x liminfν f ν(xν) gt minusinfin150
Proof We proceed by contradiction Suppose that xν isin Dν rarr x and151
liminfν f ν(xν) = minusinfin By assumption f gt minusinfin on D thus the xν cannot converge152
to a point in D ie necessarily x isin D If thats the case and since epi f ν rarr epi f 153
the line x times R would have to lie in epi f contradicting the assumption that f the154
epi-limit of the f ν belongs to f v-fcn(Rn) 155
123
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Example 1 [an epi-limit thats not in f v-fcn(Rn)] Consider the sequence of functions156 f ν [0infin) rarr R
νisinIN with157
f ν(x) =
⎧⎪⎨
⎪⎩
minusν2x if 0 le x le νminus1
ν2x minus 2ν if νminus1 le x le 2νminus1
0 for x ge 2νminus1
158
Detail The functions f ν isin f v-fcn(R) and for the sequence xν = νminus1 f ν(xν) rarr minusinfin159
and f ν rarre f where f [0infin) rarr R with f equiv 0 on (0infin) and f (0) = minusinfin Thus the160
functions f ν epi-converge to f as functions in fcn(R) provided they are appropriately161
extended ie taking on the value infin on (minusinfin 0) But they do not epi-converge to a162
function in f v-fcn(R) 163
In addition to the lsquogeometricrsquo definition the next proposition provides an lsquoanalyticrsquo164
characterization of epi-converging sequences in f v-fcn(Rn)165
Proposition 2 [epi-convergence in f v-fcn(Rn)] Let
f D rarr R f ν Dν rarr166
R ν isin IN
be a collection of functions in f v-fcn(Rn) Then f ν rarre f if and only the167
following conditions are satisfied168
(a) forall xν isin Dν rarr x in D liminfν f ν(xν) ge f (x)169
(ainfin) for all xν isin Dν rarr x isin D f ν(xν)infin4170
(b) forallx isin D exist xν isin Dν rarr x such that limsupν f ν(xν) le f (x)171
Proof If epi f ν rarr epi f and xν isin Dν rarr x either lim infν f ν(xν) = α lt infin or172
not Lemma 1 reminds us that α = minusinfin is not a possibility In the first instance173
(x α) is a cluster point of(xν f ν(xν)) isin epi f ν
νisinIN and thus belongs to epi f 174
ie f (x) le α and hence (a) holds α gt minusinfin since otherwise f would not be finite175
valued on D If α = infin that means that f ν(xν)infin and x cannot belong to D176
and thus (ainfin) holds On the other hand if x isin D and thus f (x) is finite there is177
a(xν αν) isin epi f ν
νisinIN such that xν isin Dν rarr x isin D and αν rarr f (x) with178
αν ge f ν(xν) ie lim supν f ν(xν) le f (x) ie (b) is also satisfied179
Conversely if (a) and (ainfin) hold and (xν αν) isin epi f ν rarr (x α) then either x isin D180
or not recall also that in view of Lemma 1 α cannot be minusinfin since we are dealing181
with epi-convergence in f v-fcn(Rn) In the latter instance by (ainfin) α = infin so we are182
not dealing with a converging sequence of points (in Rn+1) and there is no need to183
consider this situation any further When x isin D since then lim infν f ν(xν) ge f (x)184
and αν ge f ν(xν) one has α ge f (x) and consequently (x α) belongs to epi f 185
this means that condition (a-set) is satisfied If (x α) isin epi f from (b) follows the186
existence of a sequence xν isin Dν rarr x such that lim supν f ν(xν) le f (x) le α We187
can then choose the αν ge f ν(xν) so that αν rarr α that yields (b-set) the second188
condition for the set-convergence of epi f ν rarr epi f 189
Theorem 1 (epi-convergence basic properties) Consider a sequence
f ν Dν rarr190
R ν isin IN sub f v-fcn(Rn) epi-converging to f D rarr R also in f v-fcn(Rn) Then191
4 means non-decreasing and converging to ie not necessarily monotonically
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
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Variational convergence of bivariate functions
lim supνrarrinfin
(inf f ν) le inf f192
Moreover if xk isin argminDνk f νk for some subsequence νk and xk rarr x then193
x isin argminD f and minDνk f νk rarr minD f 194
If argminD f is a singleton then every convergent subsequence of minimizers con-195
verges to argminD f 196
Proof Let xlinfinl=1 be a sequence in D such that f (xl) rarr inf f By 2(b) for each l197
one can find a sequence xνl isin Dν rarr xl such that lim supν f ν(xνl) le f (xl) Since198
for all ν inf f ν le f ν(xνl) it follows that for all l199
lim supν
(inf f ν) le lim supν
f ν(xνl) le f (xl)200
and one has lim supν(inf f ν) le inf f since f (xl) rarr inf f 201
For the sequence xk isin Dνk rarr x from the above and 2(a)202
inf f ge lim supk
f νk (xk) ge lim infk
f νk (xk) ge f (x)203
ie x minimizes f on D and f νk (xk) = minDνk f νk rarr minD f 204
Finally since every convergent subsequence of minimizers of the functions f ν205
converges to a minimizer of f it follows that it must converge to the unique minimizer206
when argminD f is a singleton 207
In most of the applications we shall rely on a somewhat more restrictive notion208
than lsquosimplersquo epi-convergence to guarantee the convergence of the infimums209
Definition 2 (tight epi-convergence) The sequence210
f ν Dν rarr RνisinIN sub f v-fcn(Rn) epi-converges tightly to f D rarr R isin211
f v-fcn(Rn) if f ν rarre f and for all ε gt 0 there exist a compact set Bε and an in-212
dex νε such that213
forall ν ge νε inf BεcapDν f ν le inf Dν f ν + ε214
Theorem 2 (convergence of the infimums) Let f ν Dν rarr RνisinIN sub f v-fcn(Rn)215
be a sequence of functions that epi-converges to the function f D rarr R also in216
f v-fcn(Rn) with inf D f finite Then they epi-converge tightly217
(a) if and only if inf Dν f ν rarr inf D f 218
(b) if and only if there exists a sequence εν 0 such that εν-argmin f ν rarr argmin f 219
Proof Lets start with necessity in (a) For given ε gt 0 the assumptions and Theorem 1220
imply221
lim infν
( infDνcapBε
f ν) le lim infν
(infDν
f ν) + ε le lim supν
(infDν
f ν) + ε le infD
f + ε lt infin222
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223
follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224
point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225
with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226
it follows that there is no such sequences with κ arbitrarily negative In other words227
excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228
from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229
a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230
Hence231
inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232
In combination with our first string of inequalities and the fact that ε gt 0 can be233
chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234
Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235
enough given any δ gt 0 to exhibit a compact set B such that lim supν
(inf BcapDν f ν
) le236
inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237
f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238
Let B be any compact set large enough to contain all the points xν Then inf B f ν le239
f ν(xν) for all ν so B has the desired property240
We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241
tion and consequently for ν large enough also αν is finite Since convergence of242
the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243
find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244
εν = αν minus αν 245
For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246
f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247
because f ν rarre f one obtains248
inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249
le lim infν(inf f ν) le limsupν(inf f ν) le inf f250
where the last inequality comes from Theorem 1 251
Remark 1 (convergence of domains) Although epi-convergence essentially implies252
convergence of the level sets [14 Proposition 77] it does not follow that it implies253
the convergence of their (effective) domains Indeed consider the following sequence254
f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255
function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256
This vigorously argues against the temptation of involving the convergence of their257
domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258
This concludes the presentation of the results that will be used in the sequel As259
indicated earlier its also possible to derive these results from those for extended real-260
valued functions To do so one identifies f v-fcn(Rn) with261
pr -fcn(Rn) = f isin fcn(Rn)
∣∣ minus infin lt f equiv infin
262
123
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f
Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
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η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
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Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
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Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
points of non-cooperative games equilibrium points of zero-sum games etc We make33
a distinction between the situations when the bivariate function is generated from a34
single-valued mapping [11] or when the mapping can also be set-valued [10]35
The major tool is the notion of lopsided convergence that was introduced in [2]36
but is modified here so that a wider class of applications can be handled The major37
adjustment is that bivariate functions are not as in [2] no longer defined on all of38
Rn timesR
m with values in the extended reals but are now only finite-valued on a specific39
product C times D with C D subsets of Rn and R
m Dealing with lsquogeneralrsquo bivariate40
functions defined on the full product space was in keeping with the elegant work41
of Rockafellar [13] on duality relations for convexndashconcave bivariate functions and42
the subsequent work [3] on the epihypo-convergence of saddle functions However43
our present analysis actually shows that notwithstanding its esthetic allurement one44
should not cast bivariate functions even in the convexndashconcave case in the general45
extended-real valued framework In some way this is in contradiction with the uni-46
variate case where the extension by allowing for the values plusmninfin of functions defined47
on a (constrained) set to all of Rn has been so effectively exploited to derive a lsquounifiedrsquo48
convergence and differentiation theory [514] We shall show that some of this can be49
recovered but one must first make a clear distinction between max-inf problems and50
min-sup ones and only then one can generate the appropriate extensions after all51
also in the univariate case one makes a clear distinction when extending a function in52
a minimization setting or a maximization setting53
In order to be consistent in our presentation and to set up the results required54
later on we begin by a presentation of the theory of epi-convergence for real-valued55
univariate functions that are only defined on a subset of Rn No lsquonewrsquo results are56
123
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Variational convergence of bivariate functions
actually derived although a revised formulation is required We make the connection57
with the standard approach ie when these (univariate) functions are extended real-58
valued We then turn to lopsided convergence and point out the shortcomings of an59
lsquoextended real-valuedrsquo approach Finally we exploit our convergence result to obtain60
a extension of Ky Fan inequality [7] to situations when the domain of definition of the61
bivariate function is not necessarily compact62
2 Epi-convergence63
One can always represent an optimization problem involving constraints or not as64
one of minimizing an extended real-valued function In the case of a constrained-65
minimization problem simply redefine the objective as taking on the value infin out-66
side the feasible region the set determined by the constraints In this framework the67
canonical problem can be formulated as one of minimizing on all of Rn an extended68
real-valued function f Rn rarr R Approximation issues can consequently be stud-69
ied in terms of the convergence of such functions This has lead to the notion of70
epi-convergence1 that plays a key role in lsquoVariational Analysisrsquo [1514] when deal-71
ing with a maximization problem it is hypo-convergence the convergence of the72
hypographs that is the appropriate convergence notion73
Henceforth we restrict our development to the lsquominimization settingrsquo but at the74
end of this section we translate results and observations to the lsquomaximizationrsquo case75
As already indicated in Variational Analysis one usually deals with76
fcn(Rn) = f R
n rarr R
77
the space of extended real-valued functions that are defined on all of Rn even allowing78
for the possibility that they are nowhere finite-valued Definitions properties limits79
etc generally do not refer to the domain on which they are finite For reasons that will80
become clearer when we deal with the convergence of bivariate functions we need to81
depart from this simple and very convenient paradigm Our focus will be on82
f v-fcn(Rn) = f D rarr R
∣∣ for some empty = D sub R
n83
the class of all finite-valued functions with non-empty domain D sub Rn It must be84
understood that in this notation Rn does not refer to the domain of definition but to85
the underlying space that contains the domains on which the functions are defined86
The epigraph of a function f is always the set of all points in Rn+1 that lie on87
or above the graph of f irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) If88
f D rarr R belongs to f v-fcn(Rn) then89
epi f = (x α) isin D times R
∣∣ α ge f (x)
sub Rn+190
1 For extensive references and a survey of the field one can consult [15] and in particular the Commentarysection that concludes [14 Chap 7]
123
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A Jofreacute R J -B Wets
and if f belongs to fcn(Rn) then91
epi f = (x α) isin R
n+1∣∣α ge f (x)
92
A function f is lsc (=lower semicontinuous) if its epigraph is closed as a subset of93
Rn+1 ie epi f = cl(epi f ) with cl denoting closure [14 Theorem 16]294
So when f isin f v-fcn(Rn) lsc implies3 that for all xν isin D rarr x 95
ndash if x isin D liminfν f (xν) ge f (x) and96
ndash if x isin cl D D f (xν) rarr infin97
In our lsquominimizationrsquo framework cl f denotes the function whose epigraph is the98
closure relative to Rn+1 of the epigraph of f ie the lsc-regularization of f Its99
possible that when f isin f v-fcn cl f might be defined on a set thats strictly larger than100
D but always contained in cl D101
Lets now turn to convergence issues Recall that set-convergence in the Painleveacutendash102
Kuratowski sense [14 Sect 4B] is defined as follows Cν rarr C sub Rn if103
ndash (a-set) all cluster points of a sequence
xν isin CννisinIN belong to C 104
ndash (b-set) for each x isin C one can find a sequence xν isin Cν rarr x 105
When just condition (a-set) holds then C is then the outer limit of the sequence106 Cν
νisinIN and when its just (b-set) that holds then C is the inner limit [14 Chap 4107
Sect 2] Note that whenever C is the limit the outer- or the inner-limit its closed108
[14 Proposition 44] and that C = empty if and only if the sequence Cν eventually lsquoes-109
capesrsquo from any bounded set [14 Corollary 411] Moreover if the sequenceCν
νisinIN110
consists of convex sets its inner limit and its limit if it exists are also convex [14111
Proposition 415]112
Definition 1 (epi-convergence) A sequence of functions
f ν ν isin IN whose113
domains lie in Rn epi-converges to a function f when epi f ν rarr epi f as subsets114
of Rn+1 again irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) One then writes115
f ν rarre f 116
Figure 1 provides an example of two functions f and f ν that are close to each117
other in terms of the distance between their epigraphsmdashie the distance between the118
location of the two jumpsmdashbut are pretty far from each other pointwise or with respect119
to the infin-normmdashie the size of the jumps120
Let f ννisinIN be a sequence of functions with domains in Rn When121
ndash epi f is the outer limit of epi f ννisinIN one refers to f as the lower epi-limit of the122
functions f ν 123
ndash epi f is the inner limit of the epi f ν one refers to f as the upper epi-limit of the124
functions f ν 125
Of course f is the epi-limit of the sequence if its both the lower and upper epi-limit126
2 Throughout its implicitly assumed that Rn is equipped with its usual Euclidiean topology
3 Indeed if liminfν f (xν) lt infin then for some subsequence νk f (xνk ) rarr α isin R and because epi f isclosed it implies that (x α) isin epi f which would place x in the domain of f contradicting x isin D
123
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Variational convergence of bivariate functions
x
f
f
Fig 1 f and f ν epigraphically close to each other
Proposition 1 (properties of epi-limits) Let f ννisinIN be a sequence of functions with127
domains in Rn Then the lower and upper epi-limits and the epi-limit if it exists are128
all lsc Moreover if the functions f ν are convex so is the upper epi-limit and the129
epi-limit if it exists130
Proof Follows immediately from the properties of set-limits 131
The last proposition implies in particular that the family of lsc functions is closed132
under epi-convergence133
The definition of epi-convergence for families of functions in fcn(Rn) is the usual134
one [14 Chap 7 Sect B] with all the implications concerning the convergence of135
the minimizers and infimal values [14 Chap 7 Sect E] But in a certain sense the136
definition is lsquonewrsquo when the focus is on epi-convergent families in f v-fcn(Rn) and137
its for this class of functions that we need to know the conditions under which one can138
claim convergence of the minimizers and infimums We chose to make the presentation139
self-contained although as will be shown later one could also embed f v-fcn(Rn) in140
a subclass of fcn(Rn) and then appeal to the lsquostandardrsquo results but unfortunately this141
requires that the non-initiated reader plows through a substantial amount of material142
When f is an epi-limit its necessarily a lsc function since its epigraph is the set-143
limit of a collection of sets in Rn+1 Its epigraph is closed but its domain D is not144
necessarily closed Simply think of the collection of functions f ν = f for all ν with145
D = (0infin) and f (x) = 1x on D This collection clearly epi-converges to the lsc146
function f on D with closed epigraph but not with closed domain147
Lemma 1 (epi-limit value at boundary points) Suppose f D rarr R is the epi-limit148
of a sequence
f ν Dν rarr RνisinIN with all functions in f v-fcn(Rn) Then for any149
sequence xν isin Dν rarr x liminfν f ν(xν) gt minusinfin150
Proof We proceed by contradiction Suppose that xν isin Dν rarr x and151
liminfν f ν(xν) = minusinfin By assumption f gt minusinfin on D thus the xν cannot converge152
to a point in D ie necessarily x isin D If thats the case and since epi f ν rarr epi f 153
the line x times R would have to lie in epi f contradicting the assumption that f the154
epi-limit of the f ν belongs to f v-fcn(Rn) 155
123
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Example 1 [an epi-limit thats not in f v-fcn(Rn)] Consider the sequence of functions156 f ν [0infin) rarr R
νisinIN with157
f ν(x) =
⎧⎪⎨
⎪⎩
minusν2x if 0 le x le νminus1
ν2x minus 2ν if νminus1 le x le 2νminus1
0 for x ge 2νminus1
158
Detail The functions f ν isin f v-fcn(R) and for the sequence xν = νminus1 f ν(xν) rarr minusinfin159
and f ν rarre f where f [0infin) rarr R with f equiv 0 on (0infin) and f (0) = minusinfin Thus the160
functions f ν epi-converge to f as functions in fcn(R) provided they are appropriately161
extended ie taking on the value infin on (minusinfin 0) But they do not epi-converge to a162
function in f v-fcn(R) 163
In addition to the lsquogeometricrsquo definition the next proposition provides an lsquoanalyticrsquo164
characterization of epi-converging sequences in f v-fcn(Rn)165
Proposition 2 [epi-convergence in f v-fcn(Rn)] Let
f D rarr R f ν Dν rarr166
R ν isin IN
be a collection of functions in f v-fcn(Rn) Then f ν rarre f if and only the167
following conditions are satisfied168
(a) forall xν isin Dν rarr x in D liminfν f ν(xν) ge f (x)169
(ainfin) for all xν isin Dν rarr x isin D f ν(xν)infin4170
(b) forallx isin D exist xν isin Dν rarr x such that limsupν f ν(xν) le f (x)171
Proof If epi f ν rarr epi f and xν isin Dν rarr x either lim infν f ν(xν) = α lt infin or172
not Lemma 1 reminds us that α = minusinfin is not a possibility In the first instance173
(x α) is a cluster point of(xν f ν(xν)) isin epi f ν
νisinIN and thus belongs to epi f 174
ie f (x) le α and hence (a) holds α gt minusinfin since otherwise f would not be finite175
valued on D If α = infin that means that f ν(xν)infin and x cannot belong to D176
and thus (ainfin) holds On the other hand if x isin D and thus f (x) is finite there is177
a(xν αν) isin epi f ν
νisinIN such that xν isin Dν rarr x isin D and αν rarr f (x) with178
αν ge f ν(xν) ie lim supν f ν(xν) le f (x) ie (b) is also satisfied179
Conversely if (a) and (ainfin) hold and (xν αν) isin epi f ν rarr (x α) then either x isin D180
or not recall also that in view of Lemma 1 α cannot be minusinfin since we are dealing181
with epi-convergence in f v-fcn(Rn) In the latter instance by (ainfin) α = infin so we are182
not dealing with a converging sequence of points (in Rn+1) and there is no need to183
consider this situation any further When x isin D since then lim infν f ν(xν) ge f (x)184
and αν ge f ν(xν) one has α ge f (x) and consequently (x α) belongs to epi f 185
this means that condition (a-set) is satisfied If (x α) isin epi f from (b) follows the186
existence of a sequence xν isin Dν rarr x such that lim supν f ν(xν) le f (x) le α We187
can then choose the αν ge f ν(xν) so that αν rarr α that yields (b-set) the second188
condition for the set-convergence of epi f ν rarr epi f 189
Theorem 1 (epi-convergence basic properties) Consider a sequence
f ν Dν rarr190
R ν isin IN sub f v-fcn(Rn) epi-converging to f D rarr R also in f v-fcn(Rn) Then191
4 means non-decreasing and converging to ie not necessarily monotonically
123
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Variational convergence of bivariate functions
lim supνrarrinfin
(inf f ν) le inf f192
Moreover if xk isin argminDνk f νk for some subsequence νk and xk rarr x then193
x isin argminD f and minDνk f νk rarr minD f 194
If argminD f is a singleton then every convergent subsequence of minimizers con-195
verges to argminD f 196
Proof Let xlinfinl=1 be a sequence in D such that f (xl) rarr inf f By 2(b) for each l197
one can find a sequence xνl isin Dν rarr xl such that lim supν f ν(xνl) le f (xl) Since198
for all ν inf f ν le f ν(xνl) it follows that for all l199
lim supν
(inf f ν) le lim supν
f ν(xνl) le f (xl)200
and one has lim supν(inf f ν) le inf f since f (xl) rarr inf f 201
For the sequence xk isin Dνk rarr x from the above and 2(a)202
inf f ge lim supk
f νk (xk) ge lim infk
f νk (xk) ge f (x)203
ie x minimizes f on D and f νk (xk) = minDνk f νk rarr minD f 204
Finally since every convergent subsequence of minimizers of the functions f ν205
converges to a minimizer of f it follows that it must converge to the unique minimizer206
when argminD f is a singleton 207
In most of the applications we shall rely on a somewhat more restrictive notion208
than lsquosimplersquo epi-convergence to guarantee the convergence of the infimums209
Definition 2 (tight epi-convergence) The sequence210
f ν Dν rarr RνisinIN sub f v-fcn(Rn) epi-converges tightly to f D rarr R isin211
f v-fcn(Rn) if f ν rarre f and for all ε gt 0 there exist a compact set Bε and an in-212
dex νε such that213
forall ν ge νε inf BεcapDν f ν le inf Dν f ν + ε214
Theorem 2 (convergence of the infimums) Let f ν Dν rarr RνisinIN sub f v-fcn(Rn)215
be a sequence of functions that epi-converges to the function f D rarr R also in216
f v-fcn(Rn) with inf D f finite Then they epi-converge tightly217
(a) if and only if inf Dν f ν rarr inf D f 218
(b) if and only if there exists a sequence εν 0 such that εν-argmin f ν rarr argmin f 219
Proof Lets start with necessity in (a) For given ε gt 0 the assumptions and Theorem 1220
imply221
lim infν
( infDνcapBε
f ν) le lim infν
(infDν
f ν) + ε le lim supν
(infDν
f ν) + ε le infD
f + ε lt infin222
123
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If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223
follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224
point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225
with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226
it follows that there is no such sequences with κ arbitrarily negative In other words227
excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228
from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229
a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230
Hence231
inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232
In combination with our first string of inequalities and the fact that ε gt 0 can be233
chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234
Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235
enough given any δ gt 0 to exhibit a compact set B such that lim supν
(inf BcapDν f ν
) le236
inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237
f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238
Let B be any compact set large enough to contain all the points xν Then inf B f ν le239
f ν(xν) for all ν so B has the desired property240
We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241
tion and consequently for ν large enough also αν is finite Since convergence of242
the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243
find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244
εν = αν minus αν 245
For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246
f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247
because f ν rarre f one obtains248
inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249
le lim infν(inf f ν) le limsupν(inf f ν) le inf f250
where the last inequality comes from Theorem 1 251
Remark 1 (convergence of domains) Although epi-convergence essentially implies252
convergence of the level sets [14 Proposition 77] it does not follow that it implies253
the convergence of their (effective) domains Indeed consider the following sequence254
f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255
function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256
This vigorously argues against the temptation of involving the convergence of their257
domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258
This concludes the presentation of the results that will be used in the sequel As259
indicated earlier its also possible to derive these results from those for extended real-260
valued functions To do so one identifies f v-fcn(Rn) with261
pr -fcn(Rn) = f isin fcn(Rn)
∣∣ minus infin lt f equiv infin
262
123
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Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
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η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
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Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
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Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
the functions gν hypo-converge to g Maxinf-points for Fν and F are then maximizers599
of the corresponding functions gν and g The assertions now follow immediately600
from the convergence of the argmax of hypo-converging sequences cf Theorem [14601
Theorem 731] translated to the lsquomaximizationrsquo framework 602
To deal with a lsquominsuprsquo situations one can either repeat all the arguments changing603
inf to sup liminf to limsup and vice-versa or simply re-integrate the questions to604
the lsquomaxinfrsquo framework by changing signs of the approximating and limit bivariate605
functions606
123
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Variational convergence of bivariate functions
4 Ky Fanrsquos Inequality extended607
The class of usc functions is closed under hypo-converge [14 Theorem 74] and so608
is the class of concave usc functions [14 Theorem 717] A class of functions that609
is closed under lopsided convergence is the class of Ky Fan functions (Theorem 8)610
We exploit this result to obtain a generalization of Ky Fan Inequality that allows us611
to claim existence of maxinf-points in situations when the domain of definition of the612
Ky Fan function is not necessarily compact613
Definition 7 A bivariate function F C times D rarr R with Cand D convex sets in614
f v-biv(Rn+m) is called a Ky Fan function if615
(a) forall y isin D x rarr F(x y) is usc on C 616
(b) forall x isin C y rarr F(x y) is convex on D617
Note that the sets C or D are not required to be compact618
Theorem 8 (lop-limits of Ky Fan functions) The lopsided limit F C times D rarr R of a619
sequence
Fν Cν times Dν rarr RνisinIN of Ky Fan functions in f v-biv(Rn+m) is also a620
Ky Fan function621
Proof For the convexity of y rarr F(x y) let xν isin Cν rarr x isin C be the sequence set622
forth by 3(b) and y0 yλ y1 isin D with yλ = (1 minus λ)y0 + λy1 for λ isin [0 1] In view623
of 3(a) we can choose sequences
y0ν isin Dν rarr y0
y1ν isin Dν rarr y1
such that624
Fν(xν y0ν) rarr F(x y0) and Fν(xν y1ν) rarr F(x y1) Let yλν = (1 minus λ)y0ν +625
λy1ν yλν isin Dν since the functions Fν(x middot) are convex and the sequence yλννisinIN626
certainly converges to yλ For all ν one has627
Fν(xν yλν) le (1 minus λ)Fν(xν y0ν) + λFν(xν y1ν)628
Taking lininf on both sides yields629
F(x yλ) le liminfν Fν(xν yλν) le (1 minus λ)F(x y0) + λF(x y1)630
that establishes the convexity of F(x middot)631
To prove the upper semicontinuity of F with respect to x-variable we show that632
for y isin D633
hypo F(middot y) is the inner set-limit of the hypo Fν(middot yν)634
where the limit is with respect to all sequences yν isin DννisinIN converging to y and635
ν rarr infin This yields the upper semicontinuity since the inner set-limit is always636
closed and a function is usc if and only if its hypograph is closed We have to show637
that if (x α) isin hypo F(middot y) then whenever yν isin Dν rarr y one can find (xν αν) isin638
hypo Fν(middot yν) such that (xν αν) rarr (x α) But that follows immediately from 3(a)639
since we can adjust the αν le Fν(xν yν) so that they converge to α le F(x y) 640
Given a Ky Fan function with compact domain and non-negative on the diagonal641
we have the following important existence result642
123
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f
A Jofreacute R J -B Wets
Lemma 2 (Ky Fanrsquos Inequality [7] [4 Theorem 635]) Suppose F C times C rarr R is643
a Ky Fan function with C compact Then the set of maxinf-points of F is a nonempty644
subset of C Moreover if F(x x) ge 0 (on C times C) then for every maxinf-point x of645
C F(x middot) ge 0 on C646
One of the consequences of the lopsided convergence is an extension of the Ky Fanrsquos647
Inequality to the case when it is not possible to apply it directly because one of the648
conditions is not satisfied for example the compactness of the domain However we are649
able to approach the bivariate function F by a sequence FννisinIN defined on compact650
sets Cν This procedure could be useful in many situation where the original maxinf-651
problem is unbounded and then the problem is approached by a family of truncated652
maxinf-problems Such is the case for example when we consider as variables in653
the original problem the multipliers associated to inequality constraints or when the654
original problem is a Walras equilibrium with a positive orthant as consumption set655
in [8] one is precisely confronted with such situations Another simple illustrative656
example follows the statement of the theorem657
Theorem 9 (Extension of Ky Fanrsquos Inequality) Let empty = C sub Rn and F a finite-valued658
bivariate function defined on CtimesC Suppose one can find sequences of compact convex659
setsCν sub R
n
and (finite-valued) Ky Fan functions Fν Cν times Cν rarr RνisinIN lop-660
converging y-tightly to F then every cluster point x of any sequence xν ν isin IN of661
maxinf-points of the Fν is a maxinf-point of F662
Proof Ky Fanrsquos Inequality 2 implies that for all ν the set of maxinf-points of Fν is663
non-empty On the other hand in view of Theorems 8 and 4 any cluster point of such664
maxinf-points will be a maxinf-point of F 665
Example 2 (Extended Ky Fanrsquos Inequality applied) We consider a Ky Fan function666
F(x y) = sin x + (y + 1)minus1 defined on the set [0infin)2 Although667
inf yisin[0infin) F(x y) = sin x668
and the set maxinf-points is not empty we cannot apply Ky Fan Inequality because669
the domain of F is not compact the function F(middot y) is not even sup-compact670
Detail If we consider the functions Fν(x y) = sin x + (y + 1)minus1 on the compact671
domains [0 ν]2 one can apply Ky Fanrsquos Inequality Indeed in this case we have672
inf yisin[0ν) F(x y) = sin x + (ν + 1)minus1673
that converges pointwise and hypo- to sin x and674
argmaxyisin[0ν] inf yisin[0ν] F(x y) = π2 + 2kπ
∣∣ k isin IN
675
Thus xν = π2 and xν = π2 + 2νπ are maxinf-points of the Fν The sequence676
xννisinIN converges to a maxinf-point of F the second sequence xννisinIN does not677
678
123
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Variational convergence of bivariate functions
References679
1 Attouch H Variational convergence for functions and operators Applicable Mathematics Series680
Pitman London (1984)681
2 Attouch H Wets RB Convergence des points minsup et de points fixes C R Acad Sci682
Paris 296 657ndash660 (1983)683
3 Attouch H Wets RB A convergence theory for saddle functions Trans Am Math Soc 280684
1ndash41 (1983)685
4 Aubin JP Ekeland I Applied nonlinear analysis Wiley London (1984)686
5 Aubin JP Frankowska H Set-valued analysis Birkhaumluser (1990)687
6 Bagh A Approximation for optimal control problems Lecture at Universidad de Chile Santiago688
(1999)689
7 Fan K A minimax inequality and applications In Shisha O (ed) InequalitiesmdashIII pp 103ndash113690
Academic Dublin (1972)691
8 Jofreacute A Rockafellar R Wets RB A variational inequality scheme for determining an economic692
equilibrium of classical or extended type In Giannessi F Maugeri A (eds) Variational analysis and693
applications pp 553ndash578 Springer New York (2005)694
9 Jofreacute A Wets RB Continuity properties of Walras equilibrium points Ann Oper Res 114 229ndash695
243 (2002)696
10 Jofreacute A Wets RB Variational convergence of bivariate functions Motivating applications II697
(Manuscript) (2006)698
11 Jofreacute A Wets RB Variational convergence of bivariate functions motivating applications I699
(Manuscript) (2004)700
12 Lignola M Morgan J Convergence of marginal functions with dependent con-701
straints Optimization 23 189ndash213 (1992)702
13 Rockafellar R Convex analysis Princeton University Press Princeton (1970)703
14 Rockafellar R Wets RB Variational analysis 2nd edn Springer New York (2004)704
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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proo
f
Variational convergence of bivariate functions
actually derived although a revised formulation is required We make the connection57
with the standard approach ie when these (univariate) functions are extended real-58
valued We then turn to lopsided convergence and point out the shortcomings of an59
lsquoextended real-valuedrsquo approach Finally we exploit our convergence result to obtain60
a extension of Ky Fan inequality [7] to situations when the domain of definition of the61
bivariate function is not necessarily compact62
2 Epi-convergence63
One can always represent an optimization problem involving constraints or not as64
one of minimizing an extended real-valued function In the case of a constrained-65
minimization problem simply redefine the objective as taking on the value infin out-66
side the feasible region the set determined by the constraints In this framework the67
canonical problem can be formulated as one of minimizing on all of Rn an extended68
real-valued function f Rn rarr R Approximation issues can consequently be stud-69
ied in terms of the convergence of such functions This has lead to the notion of70
epi-convergence1 that plays a key role in lsquoVariational Analysisrsquo [1514] when deal-71
ing with a maximization problem it is hypo-convergence the convergence of the72
hypographs that is the appropriate convergence notion73
Henceforth we restrict our development to the lsquominimization settingrsquo but at the74
end of this section we translate results and observations to the lsquomaximizationrsquo case75
As already indicated in Variational Analysis one usually deals with76
fcn(Rn) = f R
n rarr R
77
the space of extended real-valued functions that are defined on all of Rn even allowing78
for the possibility that they are nowhere finite-valued Definitions properties limits79
etc generally do not refer to the domain on which they are finite For reasons that will80
become clearer when we deal with the convergence of bivariate functions we need to81
depart from this simple and very convenient paradigm Our focus will be on82
f v-fcn(Rn) = f D rarr R
∣∣ for some empty = D sub R
n83
the class of all finite-valued functions with non-empty domain D sub Rn It must be84
understood that in this notation Rn does not refer to the domain of definition but to85
the underlying space that contains the domains on which the functions are defined86
The epigraph of a function f is always the set of all points in Rn+1 that lie on87
or above the graph of f irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) If88
f D rarr R belongs to f v-fcn(Rn) then89
epi f = (x α) isin D times R
∣∣ α ge f (x)
sub Rn+190
1 For extensive references and a survey of the field one can consult [15] and in particular the Commentarysection that concludes [14 Chap 7]
123
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A Jofreacute R J -B Wets
and if f belongs to fcn(Rn) then91
epi f = (x α) isin R
n+1∣∣α ge f (x)
92
A function f is lsc (=lower semicontinuous) if its epigraph is closed as a subset of93
Rn+1 ie epi f = cl(epi f ) with cl denoting closure [14 Theorem 16]294
So when f isin f v-fcn(Rn) lsc implies3 that for all xν isin D rarr x 95
ndash if x isin D liminfν f (xν) ge f (x) and96
ndash if x isin cl D D f (xν) rarr infin97
In our lsquominimizationrsquo framework cl f denotes the function whose epigraph is the98
closure relative to Rn+1 of the epigraph of f ie the lsc-regularization of f Its99
possible that when f isin f v-fcn cl f might be defined on a set thats strictly larger than100
D but always contained in cl D101
Lets now turn to convergence issues Recall that set-convergence in the Painleveacutendash102
Kuratowski sense [14 Sect 4B] is defined as follows Cν rarr C sub Rn if103
ndash (a-set) all cluster points of a sequence
xν isin CννisinIN belong to C 104
ndash (b-set) for each x isin C one can find a sequence xν isin Cν rarr x 105
When just condition (a-set) holds then C is then the outer limit of the sequence106 Cν
νisinIN and when its just (b-set) that holds then C is the inner limit [14 Chap 4107
Sect 2] Note that whenever C is the limit the outer- or the inner-limit its closed108
[14 Proposition 44] and that C = empty if and only if the sequence Cν eventually lsquoes-109
capesrsquo from any bounded set [14 Corollary 411] Moreover if the sequenceCν
νisinIN110
consists of convex sets its inner limit and its limit if it exists are also convex [14111
Proposition 415]112
Definition 1 (epi-convergence) A sequence of functions
f ν ν isin IN whose113
domains lie in Rn epi-converges to a function f when epi f ν rarr epi f as subsets114
of Rn+1 again irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) One then writes115
f ν rarre f 116
Figure 1 provides an example of two functions f and f ν that are close to each117
other in terms of the distance between their epigraphsmdashie the distance between the118
location of the two jumpsmdashbut are pretty far from each other pointwise or with respect119
to the infin-normmdashie the size of the jumps120
Let f ννisinIN be a sequence of functions with domains in Rn When121
ndash epi f is the outer limit of epi f ννisinIN one refers to f as the lower epi-limit of the122
functions f ν 123
ndash epi f is the inner limit of the epi f ν one refers to f as the upper epi-limit of the124
functions f ν 125
Of course f is the epi-limit of the sequence if its both the lower and upper epi-limit126
2 Throughout its implicitly assumed that Rn is equipped with its usual Euclidiean topology
3 Indeed if liminfν f (xν) lt infin then for some subsequence νk f (xνk ) rarr α isin R and because epi f isclosed it implies that (x α) isin epi f which would place x in the domain of f contradicting x isin D
123
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Variational convergence of bivariate functions
x
f
f
Fig 1 f and f ν epigraphically close to each other
Proposition 1 (properties of epi-limits) Let f ννisinIN be a sequence of functions with127
domains in Rn Then the lower and upper epi-limits and the epi-limit if it exists are128
all lsc Moreover if the functions f ν are convex so is the upper epi-limit and the129
epi-limit if it exists130
Proof Follows immediately from the properties of set-limits 131
The last proposition implies in particular that the family of lsc functions is closed132
under epi-convergence133
The definition of epi-convergence for families of functions in fcn(Rn) is the usual134
one [14 Chap 7 Sect B] with all the implications concerning the convergence of135
the minimizers and infimal values [14 Chap 7 Sect E] But in a certain sense the136
definition is lsquonewrsquo when the focus is on epi-convergent families in f v-fcn(Rn) and137
its for this class of functions that we need to know the conditions under which one can138
claim convergence of the minimizers and infimums We chose to make the presentation139
self-contained although as will be shown later one could also embed f v-fcn(Rn) in140
a subclass of fcn(Rn) and then appeal to the lsquostandardrsquo results but unfortunately this141
requires that the non-initiated reader plows through a substantial amount of material142
When f is an epi-limit its necessarily a lsc function since its epigraph is the set-143
limit of a collection of sets in Rn+1 Its epigraph is closed but its domain D is not144
necessarily closed Simply think of the collection of functions f ν = f for all ν with145
D = (0infin) and f (x) = 1x on D This collection clearly epi-converges to the lsc146
function f on D with closed epigraph but not with closed domain147
Lemma 1 (epi-limit value at boundary points) Suppose f D rarr R is the epi-limit148
of a sequence
f ν Dν rarr RνisinIN with all functions in f v-fcn(Rn) Then for any149
sequence xν isin Dν rarr x liminfν f ν(xν) gt minusinfin150
Proof We proceed by contradiction Suppose that xν isin Dν rarr x and151
liminfν f ν(xν) = minusinfin By assumption f gt minusinfin on D thus the xν cannot converge152
to a point in D ie necessarily x isin D If thats the case and since epi f ν rarr epi f 153
the line x times R would have to lie in epi f contradicting the assumption that f the154
epi-limit of the f ν belongs to f v-fcn(Rn) 155
123
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Example 1 [an epi-limit thats not in f v-fcn(Rn)] Consider the sequence of functions156 f ν [0infin) rarr R
νisinIN with157
f ν(x) =
⎧⎪⎨
⎪⎩
minusν2x if 0 le x le νminus1
ν2x minus 2ν if νminus1 le x le 2νminus1
0 for x ge 2νminus1
158
Detail The functions f ν isin f v-fcn(R) and for the sequence xν = νminus1 f ν(xν) rarr minusinfin159
and f ν rarre f where f [0infin) rarr R with f equiv 0 on (0infin) and f (0) = minusinfin Thus the160
functions f ν epi-converge to f as functions in fcn(R) provided they are appropriately161
extended ie taking on the value infin on (minusinfin 0) But they do not epi-converge to a162
function in f v-fcn(R) 163
In addition to the lsquogeometricrsquo definition the next proposition provides an lsquoanalyticrsquo164
characterization of epi-converging sequences in f v-fcn(Rn)165
Proposition 2 [epi-convergence in f v-fcn(Rn)] Let
f D rarr R f ν Dν rarr166
R ν isin IN
be a collection of functions in f v-fcn(Rn) Then f ν rarre f if and only the167
following conditions are satisfied168
(a) forall xν isin Dν rarr x in D liminfν f ν(xν) ge f (x)169
(ainfin) for all xν isin Dν rarr x isin D f ν(xν)infin4170
(b) forallx isin D exist xν isin Dν rarr x such that limsupν f ν(xν) le f (x)171
Proof If epi f ν rarr epi f and xν isin Dν rarr x either lim infν f ν(xν) = α lt infin or172
not Lemma 1 reminds us that α = minusinfin is not a possibility In the first instance173
(x α) is a cluster point of(xν f ν(xν)) isin epi f ν
νisinIN and thus belongs to epi f 174
ie f (x) le α and hence (a) holds α gt minusinfin since otherwise f would not be finite175
valued on D If α = infin that means that f ν(xν)infin and x cannot belong to D176
and thus (ainfin) holds On the other hand if x isin D and thus f (x) is finite there is177
a(xν αν) isin epi f ν
νisinIN such that xν isin Dν rarr x isin D and αν rarr f (x) with178
αν ge f ν(xν) ie lim supν f ν(xν) le f (x) ie (b) is also satisfied179
Conversely if (a) and (ainfin) hold and (xν αν) isin epi f ν rarr (x α) then either x isin D180
or not recall also that in view of Lemma 1 α cannot be minusinfin since we are dealing181
with epi-convergence in f v-fcn(Rn) In the latter instance by (ainfin) α = infin so we are182
not dealing with a converging sequence of points (in Rn+1) and there is no need to183
consider this situation any further When x isin D since then lim infν f ν(xν) ge f (x)184
and αν ge f ν(xν) one has α ge f (x) and consequently (x α) belongs to epi f 185
this means that condition (a-set) is satisfied If (x α) isin epi f from (b) follows the186
existence of a sequence xν isin Dν rarr x such that lim supν f ν(xν) le f (x) le α We187
can then choose the αν ge f ν(xν) so that αν rarr α that yields (b-set) the second188
condition for the set-convergence of epi f ν rarr epi f 189
Theorem 1 (epi-convergence basic properties) Consider a sequence
f ν Dν rarr190
R ν isin IN sub f v-fcn(Rn) epi-converging to f D rarr R also in f v-fcn(Rn) Then191
4 means non-decreasing and converging to ie not necessarily monotonically
123
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Variational convergence of bivariate functions
lim supνrarrinfin
(inf f ν) le inf f192
Moreover if xk isin argminDνk f νk for some subsequence νk and xk rarr x then193
x isin argminD f and minDνk f νk rarr minD f 194
If argminD f is a singleton then every convergent subsequence of minimizers con-195
verges to argminD f 196
Proof Let xlinfinl=1 be a sequence in D such that f (xl) rarr inf f By 2(b) for each l197
one can find a sequence xνl isin Dν rarr xl such that lim supν f ν(xνl) le f (xl) Since198
for all ν inf f ν le f ν(xνl) it follows that for all l199
lim supν
(inf f ν) le lim supν
f ν(xνl) le f (xl)200
and one has lim supν(inf f ν) le inf f since f (xl) rarr inf f 201
For the sequence xk isin Dνk rarr x from the above and 2(a)202
inf f ge lim supk
f νk (xk) ge lim infk
f νk (xk) ge f (x)203
ie x minimizes f on D and f νk (xk) = minDνk f νk rarr minD f 204
Finally since every convergent subsequence of minimizers of the functions f ν205
converges to a minimizer of f it follows that it must converge to the unique minimizer206
when argminD f is a singleton 207
In most of the applications we shall rely on a somewhat more restrictive notion208
than lsquosimplersquo epi-convergence to guarantee the convergence of the infimums209
Definition 2 (tight epi-convergence) The sequence210
f ν Dν rarr RνisinIN sub f v-fcn(Rn) epi-converges tightly to f D rarr R isin211
f v-fcn(Rn) if f ν rarre f and for all ε gt 0 there exist a compact set Bε and an in-212
dex νε such that213
forall ν ge νε inf BεcapDν f ν le inf Dν f ν + ε214
Theorem 2 (convergence of the infimums) Let f ν Dν rarr RνisinIN sub f v-fcn(Rn)215
be a sequence of functions that epi-converges to the function f D rarr R also in216
f v-fcn(Rn) with inf D f finite Then they epi-converge tightly217
(a) if and only if inf Dν f ν rarr inf D f 218
(b) if and only if there exists a sequence εν 0 such that εν-argmin f ν rarr argmin f 219
Proof Lets start with necessity in (a) For given ε gt 0 the assumptions and Theorem 1220
imply221
lim infν
( infDνcapBε
f ν) le lim infν
(infDν
f ν) + ε le lim supν
(infDν
f ν) + ε le infD
f + ε lt infin222
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223
follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224
point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225
with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226
it follows that there is no such sequences with κ arbitrarily negative In other words227
excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228
from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229
a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230
Hence231
inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232
In combination with our first string of inequalities and the fact that ε gt 0 can be233
chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234
Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235
enough given any δ gt 0 to exhibit a compact set B such that lim supν
(inf BcapDν f ν
) le236
inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237
f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238
Let B be any compact set large enough to contain all the points xν Then inf B f ν le239
f ν(xν) for all ν so B has the desired property240
We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241
tion and consequently for ν large enough also αν is finite Since convergence of242
the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243
find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244
εν = αν minus αν 245
For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246
f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247
because f ν rarre f one obtains248
inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249
le lim infν(inf f ν) le limsupν(inf f ν) le inf f250
where the last inequality comes from Theorem 1 251
Remark 1 (convergence of domains) Although epi-convergence essentially implies252
convergence of the level sets [14 Proposition 77] it does not follow that it implies253
the convergence of their (effective) domains Indeed consider the following sequence254
f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255
function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256
This vigorously argues against the temptation of involving the convergence of their257
domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258
This concludes the presentation of the results that will be used in the sequel As259
indicated earlier its also possible to derive these results from those for extended real-260
valued functions To do so one identifies f v-fcn(Rn) with261
pr -fcn(Rn) = f isin fcn(Rn)
∣∣ minus infin lt f equiv infin
262
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
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Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
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Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
the functions gν hypo-converge to g Maxinf-points for Fν and F are then maximizers599
of the corresponding functions gν and g The assertions now follow immediately600
from the convergence of the argmax of hypo-converging sequences cf Theorem [14601
Theorem 731] translated to the lsquomaximizationrsquo framework 602
To deal with a lsquominsuprsquo situations one can either repeat all the arguments changing603
inf to sup liminf to limsup and vice-versa or simply re-integrate the questions to604
the lsquomaxinfrsquo framework by changing signs of the approximating and limit bivariate605
functions606
123
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Variational convergence of bivariate functions
4 Ky Fanrsquos Inequality extended607
The class of usc functions is closed under hypo-converge [14 Theorem 74] and so608
is the class of concave usc functions [14 Theorem 717] A class of functions that609
is closed under lopsided convergence is the class of Ky Fan functions (Theorem 8)610
We exploit this result to obtain a generalization of Ky Fan Inequality that allows us611
to claim existence of maxinf-points in situations when the domain of definition of the612
Ky Fan function is not necessarily compact613
Definition 7 A bivariate function F C times D rarr R with Cand D convex sets in614
f v-biv(Rn+m) is called a Ky Fan function if615
(a) forall y isin D x rarr F(x y) is usc on C 616
(b) forall x isin C y rarr F(x y) is convex on D617
Note that the sets C or D are not required to be compact618
Theorem 8 (lop-limits of Ky Fan functions) The lopsided limit F C times D rarr R of a619
sequence
Fν Cν times Dν rarr RνisinIN of Ky Fan functions in f v-biv(Rn+m) is also a620
Ky Fan function621
Proof For the convexity of y rarr F(x y) let xν isin Cν rarr x isin C be the sequence set622
forth by 3(b) and y0 yλ y1 isin D with yλ = (1 minus λ)y0 + λy1 for λ isin [0 1] In view623
of 3(a) we can choose sequences
y0ν isin Dν rarr y0
y1ν isin Dν rarr y1
such that624
Fν(xν y0ν) rarr F(x y0) and Fν(xν y1ν) rarr F(x y1) Let yλν = (1 minus λ)y0ν +625
λy1ν yλν isin Dν since the functions Fν(x middot) are convex and the sequence yλννisinIN626
certainly converges to yλ For all ν one has627
Fν(xν yλν) le (1 minus λ)Fν(xν y0ν) + λFν(xν y1ν)628
Taking lininf on both sides yields629
F(x yλ) le liminfν Fν(xν yλν) le (1 minus λ)F(x y0) + λF(x y1)630
that establishes the convexity of F(x middot)631
To prove the upper semicontinuity of F with respect to x-variable we show that632
for y isin D633
hypo F(middot y) is the inner set-limit of the hypo Fν(middot yν)634
where the limit is with respect to all sequences yν isin DννisinIN converging to y and635
ν rarr infin This yields the upper semicontinuity since the inner set-limit is always636
closed and a function is usc if and only if its hypograph is closed We have to show637
that if (x α) isin hypo F(middot y) then whenever yν isin Dν rarr y one can find (xν αν) isin638
hypo Fν(middot yν) such that (xν αν) rarr (x α) But that follows immediately from 3(a)639
since we can adjust the αν le Fν(xν yν) so that they converge to α le F(x y) 640
Given a Ky Fan function with compact domain and non-negative on the diagonal641
we have the following important existence result642
123
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Lemma 2 (Ky Fanrsquos Inequality [7] [4 Theorem 635]) Suppose F C times C rarr R is643
a Ky Fan function with C compact Then the set of maxinf-points of F is a nonempty644
subset of C Moreover if F(x x) ge 0 (on C times C) then for every maxinf-point x of645
C F(x middot) ge 0 on C646
One of the consequences of the lopsided convergence is an extension of the Ky Fanrsquos647
Inequality to the case when it is not possible to apply it directly because one of the648
conditions is not satisfied for example the compactness of the domain However we are649
able to approach the bivariate function F by a sequence FννisinIN defined on compact650
sets Cν This procedure could be useful in many situation where the original maxinf-651
problem is unbounded and then the problem is approached by a family of truncated652
maxinf-problems Such is the case for example when we consider as variables in653
the original problem the multipliers associated to inequality constraints or when the654
original problem is a Walras equilibrium with a positive orthant as consumption set655
in [8] one is precisely confronted with such situations Another simple illustrative656
example follows the statement of the theorem657
Theorem 9 (Extension of Ky Fanrsquos Inequality) Let empty = C sub Rn and F a finite-valued658
bivariate function defined on CtimesC Suppose one can find sequences of compact convex659
setsCν sub R
n
and (finite-valued) Ky Fan functions Fν Cν times Cν rarr RνisinIN lop-660
converging y-tightly to F then every cluster point x of any sequence xν ν isin IN of661
maxinf-points of the Fν is a maxinf-point of F662
Proof Ky Fanrsquos Inequality 2 implies that for all ν the set of maxinf-points of Fν is663
non-empty On the other hand in view of Theorems 8 and 4 any cluster point of such664
maxinf-points will be a maxinf-point of F 665
Example 2 (Extended Ky Fanrsquos Inequality applied) We consider a Ky Fan function666
F(x y) = sin x + (y + 1)minus1 defined on the set [0infin)2 Although667
inf yisin[0infin) F(x y) = sin x668
and the set maxinf-points is not empty we cannot apply Ky Fan Inequality because669
the domain of F is not compact the function F(middot y) is not even sup-compact670
Detail If we consider the functions Fν(x y) = sin x + (y + 1)minus1 on the compact671
domains [0 ν]2 one can apply Ky Fanrsquos Inequality Indeed in this case we have672
inf yisin[0ν) F(x y) = sin x + (ν + 1)minus1673
that converges pointwise and hypo- to sin x and674
argmaxyisin[0ν] inf yisin[0ν] F(x y) = π2 + 2kπ
∣∣ k isin IN
675
Thus xν = π2 and xν = π2 + 2νπ are maxinf-points of the Fν The sequence676
xννisinIN converges to a maxinf-point of F the second sequence xννisinIN does not677
678
123
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References679
1 Attouch H Variational convergence for functions and operators Applicable Mathematics Series680
Pitman London (1984)681
2 Attouch H Wets RB Convergence des points minsup et de points fixes C R Acad Sci682
Paris 296 657ndash660 (1983)683
3 Attouch H Wets RB A convergence theory for saddle functions Trans Am Math Soc 280684
1ndash41 (1983)685
4 Aubin JP Ekeland I Applied nonlinear analysis Wiley London (1984)686
5 Aubin JP Frankowska H Set-valued analysis Birkhaumluser (1990)687
6 Bagh A Approximation for optimal control problems Lecture at Universidad de Chile Santiago688
(1999)689
7 Fan K A minimax inequality and applications In Shisha O (ed) InequalitiesmdashIII pp 103ndash113690
Academic Dublin (1972)691
8 Jofreacute A Rockafellar R Wets RB A variational inequality scheme for determining an economic692
equilibrium of classical or extended type In Giannessi F Maugeri A (eds) Variational analysis and693
applications pp 553ndash578 Springer New York (2005)694
9 Jofreacute A Wets RB Continuity properties of Walras equilibrium points Ann Oper Res 114 229ndash695
243 (2002)696
10 Jofreacute A Wets RB Variational convergence of bivariate functions Motivating applications II697
(Manuscript) (2006)698
11 Jofreacute A Wets RB Variational convergence of bivariate functions motivating applications I699
(Manuscript) (2004)700
12 Lignola M Morgan J Convergence of marginal functions with dependent con-701
straints Optimization 23 189ndash213 (1992)702
13 Rockafellar R Convex analysis Princeton University Press Princeton (1970)703
14 Rockafellar R Wets RB Variational analysis 2nd edn Springer New York (2004)704
123
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A Jofreacute R J -B Wets
and if f belongs to fcn(Rn) then91
epi f = (x α) isin R
n+1∣∣α ge f (x)
92
A function f is lsc (=lower semicontinuous) if its epigraph is closed as a subset of93
Rn+1 ie epi f = cl(epi f ) with cl denoting closure [14 Theorem 16]294
So when f isin f v-fcn(Rn) lsc implies3 that for all xν isin D rarr x 95
ndash if x isin D liminfν f (xν) ge f (x) and96
ndash if x isin cl D D f (xν) rarr infin97
In our lsquominimizationrsquo framework cl f denotes the function whose epigraph is the98
closure relative to Rn+1 of the epigraph of f ie the lsc-regularization of f Its99
possible that when f isin f v-fcn cl f might be defined on a set thats strictly larger than100
D but always contained in cl D101
Lets now turn to convergence issues Recall that set-convergence in the Painleveacutendash102
Kuratowski sense [14 Sect 4B] is defined as follows Cν rarr C sub Rn if103
ndash (a-set) all cluster points of a sequence
xν isin CννisinIN belong to C 104
ndash (b-set) for each x isin C one can find a sequence xν isin Cν rarr x 105
When just condition (a-set) holds then C is then the outer limit of the sequence106 Cν
νisinIN and when its just (b-set) that holds then C is the inner limit [14 Chap 4107
Sect 2] Note that whenever C is the limit the outer- or the inner-limit its closed108
[14 Proposition 44] and that C = empty if and only if the sequence Cν eventually lsquoes-109
capesrsquo from any bounded set [14 Corollary 411] Moreover if the sequenceCν
νisinIN110
consists of convex sets its inner limit and its limit if it exists are also convex [14111
Proposition 415]112
Definition 1 (epi-convergence) A sequence of functions
f ν ν isin IN whose113
domains lie in Rn epi-converges to a function f when epi f ν rarr epi f as subsets114
of Rn+1 again irrespective of f belonging to f v-fcn(Rn) or fcn(Rn) One then writes115
f ν rarre f 116
Figure 1 provides an example of two functions f and f ν that are close to each117
other in terms of the distance between their epigraphsmdashie the distance between the118
location of the two jumpsmdashbut are pretty far from each other pointwise or with respect119
to the infin-normmdashie the size of the jumps120
Let f ννisinIN be a sequence of functions with domains in Rn When121
ndash epi f is the outer limit of epi f ννisinIN one refers to f as the lower epi-limit of the122
functions f ν 123
ndash epi f is the inner limit of the epi f ν one refers to f as the upper epi-limit of the124
functions f ν 125
Of course f is the epi-limit of the sequence if its both the lower and upper epi-limit126
2 Throughout its implicitly assumed that Rn is equipped with its usual Euclidiean topology
3 Indeed if liminfν f (xν) lt infin then for some subsequence νk f (xνk ) rarr α isin R and because epi f isclosed it implies that (x α) isin epi f which would place x in the domain of f contradicting x isin D
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functions
x
f
f
Fig 1 f and f ν epigraphically close to each other
Proposition 1 (properties of epi-limits) Let f ννisinIN be a sequence of functions with127
domains in Rn Then the lower and upper epi-limits and the epi-limit if it exists are128
all lsc Moreover if the functions f ν are convex so is the upper epi-limit and the129
epi-limit if it exists130
Proof Follows immediately from the properties of set-limits 131
The last proposition implies in particular that the family of lsc functions is closed132
under epi-convergence133
The definition of epi-convergence for families of functions in fcn(Rn) is the usual134
one [14 Chap 7 Sect B] with all the implications concerning the convergence of135
the minimizers and infimal values [14 Chap 7 Sect E] But in a certain sense the136
definition is lsquonewrsquo when the focus is on epi-convergent families in f v-fcn(Rn) and137
its for this class of functions that we need to know the conditions under which one can138
claim convergence of the minimizers and infimums We chose to make the presentation139
self-contained although as will be shown later one could also embed f v-fcn(Rn) in140
a subclass of fcn(Rn) and then appeal to the lsquostandardrsquo results but unfortunately this141
requires that the non-initiated reader plows through a substantial amount of material142
When f is an epi-limit its necessarily a lsc function since its epigraph is the set-143
limit of a collection of sets in Rn+1 Its epigraph is closed but its domain D is not144
necessarily closed Simply think of the collection of functions f ν = f for all ν with145
D = (0infin) and f (x) = 1x on D This collection clearly epi-converges to the lsc146
function f on D with closed epigraph but not with closed domain147
Lemma 1 (epi-limit value at boundary points) Suppose f D rarr R is the epi-limit148
of a sequence
f ν Dν rarr RνisinIN with all functions in f v-fcn(Rn) Then for any149
sequence xν isin Dν rarr x liminfν f ν(xν) gt minusinfin150
Proof We proceed by contradiction Suppose that xν isin Dν rarr x and151
liminfν f ν(xν) = minusinfin By assumption f gt minusinfin on D thus the xν cannot converge152
to a point in D ie necessarily x isin D If thats the case and since epi f ν rarr epi f 153
the line x times R would have to lie in epi f contradicting the assumption that f the154
epi-limit of the f ν belongs to f v-fcn(Rn) 155
123
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Example 1 [an epi-limit thats not in f v-fcn(Rn)] Consider the sequence of functions156 f ν [0infin) rarr R
νisinIN with157
f ν(x) =
⎧⎪⎨
⎪⎩
minusν2x if 0 le x le νminus1
ν2x minus 2ν if νminus1 le x le 2νminus1
0 for x ge 2νminus1
158
Detail The functions f ν isin f v-fcn(R) and for the sequence xν = νminus1 f ν(xν) rarr minusinfin159
and f ν rarre f where f [0infin) rarr R with f equiv 0 on (0infin) and f (0) = minusinfin Thus the160
functions f ν epi-converge to f as functions in fcn(R) provided they are appropriately161
extended ie taking on the value infin on (minusinfin 0) But they do not epi-converge to a162
function in f v-fcn(R) 163
In addition to the lsquogeometricrsquo definition the next proposition provides an lsquoanalyticrsquo164
characterization of epi-converging sequences in f v-fcn(Rn)165
Proposition 2 [epi-convergence in f v-fcn(Rn)] Let
f D rarr R f ν Dν rarr166
R ν isin IN
be a collection of functions in f v-fcn(Rn) Then f ν rarre f if and only the167
following conditions are satisfied168
(a) forall xν isin Dν rarr x in D liminfν f ν(xν) ge f (x)169
(ainfin) for all xν isin Dν rarr x isin D f ν(xν)infin4170
(b) forallx isin D exist xν isin Dν rarr x such that limsupν f ν(xν) le f (x)171
Proof If epi f ν rarr epi f and xν isin Dν rarr x either lim infν f ν(xν) = α lt infin or172
not Lemma 1 reminds us that α = minusinfin is not a possibility In the first instance173
(x α) is a cluster point of(xν f ν(xν)) isin epi f ν
νisinIN and thus belongs to epi f 174
ie f (x) le α and hence (a) holds α gt minusinfin since otherwise f would not be finite175
valued on D If α = infin that means that f ν(xν)infin and x cannot belong to D176
and thus (ainfin) holds On the other hand if x isin D and thus f (x) is finite there is177
a(xν αν) isin epi f ν
νisinIN such that xν isin Dν rarr x isin D and αν rarr f (x) with178
αν ge f ν(xν) ie lim supν f ν(xν) le f (x) ie (b) is also satisfied179
Conversely if (a) and (ainfin) hold and (xν αν) isin epi f ν rarr (x α) then either x isin D180
or not recall also that in view of Lemma 1 α cannot be minusinfin since we are dealing181
with epi-convergence in f v-fcn(Rn) In the latter instance by (ainfin) α = infin so we are182
not dealing with a converging sequence of points (in Rn+1) and there is no need to183
consider this situation any further When x isin D since then lim infν f ν(xν) ge f (x)184
and αν ge f ν(xν) one has α ge f (x) and consequently (x α) belongs to epi f 185
this means that condition (a-set) is satisfied If (x α) isin epi f from (b) follows the186
existence of a sequence xν isin Dν rarr x such that lim supν f ν(xν) le f (x) le α We187
can then choose the αν ge f ν(xν) so that αν rarr α that yields (b-set) the second188
condition for the set-convergence of epi f ν rarr epi f 189
Theorem 1 (epi-convergence basic properties) Consider a sequence
f ν Dν rarr190
R ν isin IN sub f v-fcn(Rn) epi-converging to f D rarr R also in f v-fcn(Rn) Then191
4 means non-decreasing and converging to ie not necessarily monotonically
123
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Variational convergence of bivariate functions
lim supνrarrinfin
(inf f ν) le inf f192
Moreover if xk isin argminDνk f νk for some subsequence νk and xk rarr x then193
x isin argminD f and minDνk f νk rarr minD f 194
If argminD f is a singleton then every convergent subsequence of minimizers con-195
verges to argminD f 196
Proof Let xlinfinl=1 be a sequence in D such that f (xl) rarr inf f By 2(b) for each l197
one can find a sequence xνl isin Dν rarr xl such that lim supν f ν(xνl) le f (xl) Since198
for all ν inf f ν le f ν(xνl) it follows that for all l199
lim supν
(inf f ν) le lim supν
f ν(xνl) le f (xl)200
and one has lim supν(inf f ν) le inf f since f (xl) rarr inf f 201
For the sequence xk isin Dνk rarr x from the above and 2(a)202
inf f ge lim supk
f νk (xk) ge lim infk
f νk (xk) ge f (x)203
ie x minimizes f on D and f νk (xk) = minDνk f νk rarr minD f 204
Finally since every convergent subsequence of minimizers of the functions f ν205
converges to a minimizer of f it follows that it must converge to the unique minimizer206
when argminD f is a singleton 207
In most of the applications we shall rely on a somewhat more restrictive notion208
than lsquosimplersquo epi-convergence to guarantee the convergence of the infimums209
Definition 2 (tight epi-convergence) The sequence210
f ν Dν rarr RνisinIN sub f v-fcn(Rn) epi-converges tightly to f D rarr R isin211
f v-fcn(Rn) if f ν rarre f and for all ε gt 0 there exist a compact set Bε and an in-212
dex νε such that213
forall ν ge νε inf BεcapDν f ν le inf Dν f ν + ε214
Theorem 2 (convergence of the infimums) Let f ν Dν rarr RνisinIN sub f v-fcn(Rn)215
be a sequence of functions that epi-converges to the function f D rarr R also in216
f v-fcn(Rn) with inf D f finite Then they epi-converge tightly217
(a) if and only if inf Dν f ν rarr inf D f 218
(b) if and only if there exists a sequence εν 0 such that εν-argmin f ν rarr argmin f 219
Proof Lets start with necessity in (a) For given ε gt 0 the assumptions and Theorem 1220
imply221
lim infν
( infDνcapBε
f ν) le lim infν
(infDν
f ν) + ε le lim supν
(infDν
f ν) + ε le infD
f + ε lt infin222
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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A Jofreacute R J -B Wets
If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223
follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224
point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225
with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226
it follows that there is no such sequences with κ arbitrarily negative In other words227
excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228
from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229
a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230
Hence231
inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232
In combination with our first string of inequalities and the fact that ε gt 0 can be233
chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234
Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235
enough given any δ gt 0 to exhibit a compact set B such that lim supν
(inf BcapDν f ν
) le236
inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237
f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238
Let B be any compact set large enough to contain all the points xν Then inf B f ν le239
f ν(xν) for all ν so B has the desired property240
We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241
tion and consequently for ν large enough also αν is finite Since convergence of242
the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243
find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244
εν = αν minus αν 245
For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246
f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247
because f ν rarre f one obtains248
inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249
le lim infν(inf f ν) le limsupν(inf f ν) le inf f250
where the last inequality comes from Theorem 1 251
Remark 1 (convergence of domains) Although epi-convergence essentially implies252
convergence of the level sets [14 Proposition 77] it does not follow that it implies253
the convergence of their (effective) domains Indeed consider the following sequence254
f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255
function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256
This vigorously argues against the temptation of involving the convergence of their257
domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258
This concludes the presentation of the results that will be used in the sequel As259
indicated earlier its also possible to derive these results from those for extended real-260
valued functions To do so one identifies f v-fcn(Rn) with261
pr -fcn(Rn) = f isin fcn(Rn)
∣∣ minus infin lt f equiv infin
262
123
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Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
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η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
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Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
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Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
the functions gν hypo-converge to g Maxinf-points for Fν and F are then maximizers599
of the corresponding functions gν and g The assertions now follow immediately600
from the convergence of the argmax of hypo-converging sequences cf Theorem [14601
Theorem 731] translated to the lsquomaximizationrsquo framework 602
To deal with a lsquominsuprsquo situations one can either repeat all the arguments changing603
inf to sup liminf to limsup and vice-versa or simply re-integrate the questions to604
the lsquomaxinfrsquo framework by changing signs of the approximating and limit bivariate605
functions606
123
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Variational convergence of bivariate functions
4 Ky Fanrsquos Inequality extended607
The class of usc functions is closed under hypo-converge [14 Theorem 74] and so608
is the class of concave usc functions [14 Theorem 717] A class of functions that609
is closed under lopsided convergence is the class of Ky Fan functions (Theorem 8)610
We exploit this result to obtain a generalization of Ky Fan Inequality that allows us611
to claim existence of maxinf-points in situations when the domain of definition of the612
Ky Fan function is not necessarily compact613
Definition 7 A bivariate function F C times D rarr R with Cand D convex sets in614
f v-biv(Rn+m) is called a Ky Fan function if615
(a) forall y isin D x rarr F(x y) is usc on C 616
(b) forall x isin C y rarr F(x y) is convex on D617
Note that the sets C or D are not required to be compact618
Theorem 8 (lop-limits of Ky Fan functions) The lopsided limit F C times D rarr R of a619
sequence
Fν Cν times Dν rarr RνisinIN of Ky Fan functions in f v-biv(Rn+m) is also a620
Ky Fan function621
Proof For the convexity of y rarr F(x y) let xν isin Cν rarr x isin C be the sequence set622
forth by 3(b) and y0 yλ y1 isin D with yλ = (1 minus λ)y0 + λy1 for λ isin [0 1] In view623
of 3(a) we can choose sequences
y0ν isin Dν rarr y0
y1ν isin Dν rarr y1
such that624
Fν(xν y0ν) rarr F(x y0) and Fν(xν y1ν) rarr F(x y1) Let yλν = (1 minus λ)y0ν +625
λy1ν yλν isin Dν since the functions Fν(x middot) are convex and the sequence yλννisinIN626
certainly converges to yλ For all ν one has627
Fν(xν yλν) le (1 minus λ)Fν(xν y0ν) + λFν(xν y1ν)628
Taking lininf on both sides yields629
F(x yλ) le liminfν Fν(xν yλν) le (1 minus λ)F(x y0) + λF(x y1)630
that establishes the convexity of F(x middot)631
To prove the upper semicontinuity of F with respect to x-variable we show that632
for y isin D633
hypo F(middot y) is the inner set-limit of the hypo Fν(middot yν)634
where the limit is with respect to all sequences yν isin DννisinIN converging to y and635
ν rarr infin This yields the upper semicontinuity since the inner set-limit is always636
closed and a function is usc if and only if its hypograph is closed We have to show637
that if (x α) isin hypo F(middot y) then whenever yν isin Dν rarr y one can find (xν αν) isin638
hypo Fν(middot yν) such that (xν αν) rarr (x α) But that follows immediately from 3(a)639
since we can adjust the αν le Fν(xν yν) so that they converge to α le F(x y) 640
Given a Ky Fan function with compact domain and non-negative on the diagonal641
we have the following important existence result642
123
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Lemma 2 (Ky Fanrsquos Inequality [7] [4 Theorem 635]) Suppose F C times C rarr R is643
a Ky Fan function with C compact Then the set of maxinf-points of F is a nonempty644
subset of C Moreover if F(x x) ge 0 (on C times C) then for every maxinf-point x of645
C F(x middot) ge 0 on C646
One of the consequences of the lopsided convergence is an extension of the Ky Fanrsquos647
Inequality to the case when it is not possible to apply it directly because one of the648
conditions is not satisfied for example the compactness of the domain However we are649
able to approach the bivariate function F by a sequence FννisinIN defined on compact650
sets Cν This procedure could be useful in many situation where the original maxinf-651
problem is unbounded and then the problem is approached by a family of truncated652
maxinf-problems Such is the case for example when we consider as variables in653
the original problem the multipliers associated to inequality constraints or when the654
original problem is a Walras equilibrium with a positive orthant as consumption set655
in [8] one is precisely confronted with such situations Another simple illustrative656
example follows the statement of the theorem657
Theorem 9 (Extension of Ky Fanrsquos Inequality) Let empty = C sub Rn and F a finite-valued658
bivariate function defined on CtimesC Suppose one can find sequences of compact convex659
setsCν sub R
n
and (finite-valued) Ky Fan functions Fν Cν times Cν rarr RνisinIN lop-660
converging y-tightly to F then every cluster point x of any sequence xν ν isin IN of661
maxinf-points of the Fν is a maxinf-point of F662
Proof Ky Fanrsquos Inequality 2 implies that for all ν the set of maxinf-points of Fν is663
non-empty On the other hand in view of Theorems 8 and 4 any cluster point of such664
maxinf-points will be a maxinf-point of F 665
Example 2 (Extended Ky Fanrsquos Inequality applied) We consider a Ky Fan function666
F(x y) = sin x + (y + 1)minus1 defined on the set [0infin)2 Although667
inf yisin[0infin) F(x y) = sin x668
and the set maxinf-points is not empty we cannot apply Ky Fan Inequality because669
the domain of F is not compact the function F(middot y) is not even sup-compact670
Detail If we consider the functions Fν(x y) = sin x + (y + 1)minus1 on the compact671
domains [0 ν]2 one can apply Ky Fanrsquos Inequality Indeed in this case we have672
inf yisin[0ν) F(x y) = sin x + (ν + 1)minus1673
that converges pointwise and hypo- to sin x and674
argmaxyisin[0ν] inf yisin[0ν] F(x y) = π2 + 2kπ
∣∣ k isin IN
675
Thus xν = π2 and xν = π2 + 2νπ are maxinf-points of the Fν The sequence676
xννisinIN converges to a maxinf-point of F the second sequence xννisinIN does not677
678
123
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Variational convergence of bivariate functions
References679
1 Attouch H Variational convergence for functions and operators Applicable Mathematics Series680
Pitman London (1984)681
2 Attouch H Wets RB Convergence des points minsup et de points fixes C R Acad Sci682
Paris 296 657ndash660 (1983)683
3 Attouch H Wets RB A convergence theory for saddle functions Trans Am Math Soc 280684
1ndash41 (1983)685
4 Aubin JP Ekeland I Applied nonlinear analysis Wiley London (1984)686
5 Aubin JP Frankowska H Set-valued analysis Birkhaumluser (1990)687
6 Bagh A Approximation for optimal control problems Lecture at Universidad de Chile Santiago688
(1999)689
7 Fan K A minimax inequality and applications In Shisha O (ed) InequalitiesmdashIII pp 103ndash113690
Academic Dublin (1972)691
8 Jofreacute A Rockafellar R Wets RB A variational inequality scheme for determining an economic692
equilibrium of classical or extended type In Giannessi F Maugeri A (eds) Variational analysis and693
applications pp 553ndash578 Springer New York (2005)694
9 Jofreacute A Wets RB Continuity properties of Walras equilibrium points Ann Oper Res 114 229ndash695
243 (2002)696
10 Jofreacute A Wets RB Variational convergence of bivariate functions Motivating applications II697
(Manuscript) (2006)698
11 Jofreacute A Wets RB Variational convergence of bivariate functions motivating applications I699
(Manuscript) (2004)700
12 Lignola M Morgan J Convergence of marginal functions with dependent con-701
straints Optimization 23 189ndash213 (1992)702
13 Rockafellar R Convex analysis Princeton University Press Princeton (1970)703
14 Rockafellar R Wets RB Variational analysis 2nd edn Springer New York (2004)704
123
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unco
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Variational convergence of bivariate functions
x
f
f
Fig 1 f and f ν epigraphically close to each other
Proposition 1 (properties of epi-limits) Let f ννisinIN be a sequence of functions with127
domains in Rn Then the lower and upper epi-limits and the epi-limit if it exists are128
all lsc Moreover if the functions f ν are convex so is the upper epi-limit and the129
epi-limit if it exists130
Proof Follows immediately from the properties of set-limits 131
The last proposition implies in particular that the family of lsc functions is closed132
under epi-convergence133
The definition of epi-convergence for families of functions in fcn(Rn) is the usual134
one [14 Chap 7 Sect B] with all the implications concerning the convergence of135
the minimizers and infimal values [14 Chap 7 Sect E] But in a certain sense the136
definition is lsquonewrsquo when the focus is on epi-convergent families in f v-fcn(Rn) and137
its for this class of functions that we need to know the conditions under which one can138
claim convergence of the minimizers and infimums We chose to make the presentation139
self-contained although as will be shown later one could also embed f v-fcn(Rn) in140
a subclass of fcn(Rn) and then appeal to the lsquostandardrsquo results but unfortunately this141
requires that the non-initiated reader plows through a substantial amount of material142
When f is an epi-limit its necessarily a lsc function since its epigraph is the set-143
limit of a collection of sets in Rn+1 Its epigraph is closed but its domain D is not144
necessarily closed Simply think of the collection of functions f ν = f for all ν with145
D = (0infin) and f (x) = 1x on D This collection clearly epi-converges to the lsc146
function f on D with closed epigraph but not with closed domain147
Lemma 1 (epi-limit value at boundary points) Suppose f D rarr R is the epi-limit148
of a sequence
f ν Dν rarr RνisinIN with all functions in f v-fcn(Rn) Then for any149
sequence xν isin Dν rarr x liminfν f ν(xν) gt minusinfin150
Proof We proceed by contradiction Suppose that xν isin Dν rarr x and151
liminfν f ν(xν) = minusinfin By assumption f gt minusinfin on D thus the xν cannot converge152
to a point in D ie necessarily x isin D If thats the case and since epi f ν rarr epi f 153
the line x times R would have to lie in epi f contradicting the assumption that f the154
epi-limit of the f ν belongs to f v-fcn(Rn) 155
123
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Example 1 [an epi-limit thats not in f v-fcn(Rn)] Consider the sequence of functions156 f ν [0infin) rarr R
νisinIN with157
f ν(x) =
⎧⎪⎨
⎪⎩
minusν2x if 0 le x le νminus1
ν2x minus 2ν if νminus1 le x le 2νminus1
0 for x ge 2νminus1
158
Detail The functions f ν isin f v-fcn(R) and for the sequence xν = νminus1 f ν(xν) rarr minusinfin159
and f ν rarre f where f [0infin) rarr R with f equiv 0 on (0infin) and f (0) = minusinfin Thus the160
functions f ν epi-converge to f as functions in fcn(R) provided they are appropriately161
extended ie taking on the value infin on (minusinfin 0) But they do not epi-converge to a162
function in f v-fcn(R) 163
In addition to the lsquogeometricrsquo definition the next proposition provides an lsquoanalyticrsquo164
characterization of epi-converging sequences in f v-fcn(Rn)165
Proposition 2 [epi-convergence in f v-fcn(Rn)] Let
f D rarr R f ν Dν rarr166
R ν isin IN
be a collection of functions in f v-fcn(Rn) Then f ν rarre f if and only the167
following conditions are satisfied168
(a) forall xν isin Dν rarr x in D liminfν f ν(xν) ge f (x)169
(ainfin) for all xν isin Dν rarr x isin D f ν(xν)infin4170
(b) forallx isin D exist xν isin Dν rarr x such that limsupν f ν(xν) le f (x)171
Proof If epi f ν rarr epi f and xν isin Dν rarr x either lim infν f ν(xν) = α lt infin or172
not Lemma 1 reminds us that α = minusinfin is not a possibility In the first instance173
(x α) is a cluster point of(xν f ν(xν)) isin epi f ν
νisinIN and thus belongs to epi f 174
ie f (x) le α and hence (a) holds α gt minusinfin since otherwise f would not be finite175
valued on D If α = infin that means that f ν(xν)infin and x cannot belong to D176
and thus (ainfin) holds On the other hand if x isin D and thus f (x) is finite there is177
a(xν αν) isin epi f ν
νisinIN such that xν isin Dν rarr x isin D and αν rarr f (x) with178
αν ge f ν(xν) ie lim supν f ν(xν) le f (x) ie (b) is also satisfied179
Conversely if (a) and (ainfin) hold and (xν αν) isin epi f ν rarr (x α) then either x isin D180
or not recall also that in view of Lemma 1 α cannot be minusinfin since we are dealing181
with epi-convergence in f v-fcn(Rn) In the latter instance by (ainfin) α = infin so we are182
not dealing with a converging sequence of points (in Rn+1) and there is no need to183
consider this situation any further When x isin D since then lim infν f ν(xν) ge f (x)184
and αν ge f ν(xν) one has α ge f (x) and consequently (x α) belongs to epi f 185
this means that condition (a-set) is satisfied If (x α) isin epi f from (b) follows the186
existence of a sequence xν isin Dν rarr x such that lim supν f ν(xν) le f (x) le α We187
can then choose the αν ge f ν(xν) so that αν rarr α that yields (b-set) the second188
condition for the set-convergence of epi f ν rarr epi f 189
Theorem 1 (epi-convergence basic properties) Consider a sequence
f ν Dν rarr190
R ν isin IN sub f v-fcn(Rn) epi-converging to f D rarr R also in f v-fcn(Rn) Then191
4 means non-decreasing and converging to ie not necessarily monotonically
123
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Variational convergence of bivariate functions
lim supνrarrinfin
(inf f ν) le inf f192
Moreover if xk isin argminDνk f νk for some subsequence νk and xk rarr x then193
x isin argminD f and minDνk f νk rarr minD f 194
If argminD f is a singleton then every convergent subsequence of minimizers con-195
verges to argminD f 196
Proof Let xlinfinl=1 be a sequence in D such that f (xl) rarr inf f By 2(b) for each l197
one can find a sequence xνl isin Dν rarr xl such that lim supν f ν(xνl) le f (xl) Since198
for all ν inf f ν le f ν(xνl) it follows that for all l199
lim supν
(inf f ν) le lim supν
f ν(xνl) le f (xl)200
and one has lim supν(inf f ν) le inf f since f (xl) rarr inf f 201
For the sequence xk isin Dνk rarr x from the above and 2(a)202
inf f ge lim supk
f νk (xk) ge lim infk
f νk (xk) ge f (x)203
ie x minimizes f on D and f νk (xk) = minDνk f νk rarr minD f 204
Finally since every convergent subsequence of minimizers of the functions f ν205
converges to a minimizer of f it follows that it must converge to the unique minimizer206
when argminD f is a singleton 207
In most of the applications we shall rely on a somewhat more restrictive notion208
than lsquosimplersquo epi-convergence to guarantee the convergence of the infimums209
Definition 2 (tight epi-convergence) The sequence210
f ν Dν rarr RνisinIN sub f v-fcn(Rn) epi-converges tightly to f D rarr R isin211
f v-fcn(Rn) if f ν rarre f and for all ε gt 0 there exist a compact set Bε and an in-212
dex νε such that213
forall ν ge νε inf BεcapDν f ν le inf Dν f ν + ε214
Theorem 2 (convergence of the infimums) Let f ν Dν rarr RνisinIN sub f v-fcn(Rn)215
be a sequence of functions that epi-converges to the function f D rarr R also in216
f v-fcn(Rn) with inf D f finite Then they epi-converge tightly217
(a) if and only if inf Dν f ν rarr inf D f 218
(b) if and only if there exists a sequence εν 0 such that εν-argmin f ν rarr argmin f 219
Proof Lets start with necessity in (a) For given ε gt 0 the assumptions and Theorem 1220
imply221
lim infν
( infDνcapBε
f ν) le lim infν
(infDν
f ν) + ε le lim supν
(infDν
f ν) + ε le infD
f + ε lt infin222
123
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If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223
follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224
point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225
with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226
it follows that there is no such sequences with κ arbitrarily negative In other words227
excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228
from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229
a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230
Hence231
inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232
In combination with our first string of inequalities and the fact that ε gt 0 can be233
chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234
Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235
enough given any δ gt 0 to exhibit a compact set B such that lim supν
(inf BcapDν f ν
) le236
inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237
f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238
Let B be any compact set large enough to contain all the points xν Then inf B f ν le239
f ν(xν) for all ν so B has the desired property240
We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241
tion and consequently for ν large enough also αν is finite Since convergence of242
the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243
find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244
εν = αν minus αν 245
For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246
f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247
because f ν rarre f one obtains248
inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249
le lim infν(inf f ν) le limsupν(inf f ν) le inf f250
where the last inequality comes from Theorem 1 251
Remark 1 (convergence of domains) Although epi-convergence essentially implies252
convergence of the level sets [14 Proposition 77] it does not follow that it implies253
the convergence of their (effective) domains Indeed consider the following sequence254
f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255
function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256
This vigorously argues against the temptation of involving the convergence of their257
domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258
This concludes the presentation of the results that will be used in the sequel As259
indicated earlier its also possible to derive these results from those for extended real-260
valued functions To do so one identifies f v-fcn(Rn) with261
pr -fcn(Rn) = f isin fcn(Rn)
∣∣ minus infin lt f equiv infin
262
123
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Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
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η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
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Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
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Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
the functions gν hypo-converge to g Maxinf-points for Fν and F are then maximizers599
of the corresponding functions gν and g The assertions now follow immediately600
from the convergence of the argmax of hypo-converging sequences cf Theorem [14601
Theorem 731] translated to the lsquomaximizationrsquo framework 602
To deal with a lsquominsuprsquo situations one can either repeat all the arguments changing603
inf to sup liminf to limsup and vice-versa or simply re-integrate the questions to604
the lsquomaxinfrsquo framework by changing signs of the approximating and limit bivariate605
functions606
123
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Variational convergence of bivariate functions
4 Ky Fanrsquos Inequality extended607
The class of usc functions is closed under hypo-converge [14 Theorem 74] and so608
is the class of concave usc functions [14 Theorem 717] A class of functions that609
is closed under lopsided convergence is the class of Ky Fan functions (Theorem 8)610
We exploit this result to obtain a generalization of Ky Fan Inequality that allows us611
to claim existence of maxinf-points in situations when the domain of definition of the612
Ky Fan function is not necessarily compact613
Definition 7 A bivariate function F C times D rarr R with Cand D convex sets in614
f v-biv(Rn+m) is called a Ky Fan function if615
(a) forall y isin D x rarr F(x y) is usc on C 616
(b) forall x isin C y rarr F(x y) is convex on D617
Note that the sets C or D are not required to be compact618
Theorem 8 (lop-limits of Ky Fan functions) The lopsided limit F C times D rarr R of a619
sequence
Fν Cν times Dν rarr RνisinIN of Ky Fan functions in f v-biv(Rn+m) is also a620
Ky Fan function621
Proof For the convexity of y rarr F(x y) let xν isin Cν rarr x isin C be the sequence set622
forth by 3(b) and y0 yλ y1 isin D with yλ = (1 minus λ)y0 + λy1 for λ isin [0 1] In view623
of 3(a) we can choose sequences
y0ν isin Dν rarr y0
y1ν isin Dν rarr y1
such that624
Fν(xν y0ν) rarr F(x y0) and Fν(xν y1ν) rarr F(x y1) Let yλν = (1 minus λ)y0ν +625
λy1ν yλν isin Dν since the functions Fν(x middot) are convex and the sequence yλννisinIN626
certainly converges to yλ For all ν one has627
Fν(xν yλν) le (1 minus λ)Fν(xν y0ν) + λFν(xν y1ν)628
Taking lininf on both sides yields629
F(x yλ) le liminfν Fν(xν yλν) le (1 minus λ)F(x y0) + λF(x y1)630
that establishes the convexity of F(x middot)631
To prove the upper semicontinuity of F with respect to x-variable we show that632
for y isin D633
hypo F(middot y) is the inner set-limit of the hypo Fν(middot yν)634
where the limit is with respect to all sequences yν isin DννisinIN converging to y and635
ν rarr infin This yields the upper semicontinuity since the inner set-limit is always636
closed and a function is usc if and only if its hypograph is closed We have to show637
that if (x α) isin hypo F(middot y) then whenever yν isin Dν rarr y one can find (xν αν) isin638
hypo Fν(middot yν) such that (xν αν) rarr (x α) But that follows immediately from 3(a)639
since we can adjust the αν le Fν(xν yν) so that they converge to α le F(x y) 640
Given a Ky Fan function with compact domain and non-negative on the diagonal641
we have the following important existence result642
123
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Lemma 2 (Ky Fanrsquos Inequality [7] [4 Theorem 635]) Suppose F C times C rarr R is643
a Ky Fan function with C compact Then the set of maxinf-points of F is a nonempty644
subset of C Moreover if F(x x) ge 0 (on C times C) then for every maxinf-point x of645
C F(x middot) ge 0 on C646
One of the consequences of the lopsided convergence is an extension of the Ky Fanrsquos647
Inequality to the case when it is not possible to apply it directly because one of the648
conditions is not satisfied for example the compactness of the domain However we are649
able to approach the bivariate function F by a sequence FννisinIN defined on compact650
sets Cν This procedure could be useful in many situation where the original maxinf-651
problem is unbounded and then the problem is approached by a family of truncated652
maxinf-problems Such is the case for example when we consider as variables in653
the original problem the multipliers associated to inequality constraints or when the654
original problem is a Walras equilibrium with a positive orthant as consumption set655
in [8] one is precisely confronted with such situations Another simple illustrative656
example follows the statement of the theorem657
Theorem 9 (Extension of Ky Fanrsquos Inequality) Let empty = C sub Rn and F a finite-valued658
bivariate function defined on CtimesC Suppose one can find sequences of compact convex659
setsCν sub R
n
and (finite-valued) Ky Fan functions Fν Cν times Cν rarr RνisinIN lop-660
converging y-tightly to F then every cluster point x of any sequence xν ν isin IN of661
maxinf-points of the Fν is a maxinf-point of F662
Proof Ky Fanrsquos Inequality 2 implies that for all ν the set of maxinf-points of Fν is663
non-empty On the other hand in view of Theorems 8 and 4 any cluster point of such664
maxinf-points will be a maxinf-point of F 665
Example 2 (Extended Ky Fanrsquos Inequality applied) We consider a Ky Fan function666
F(x y) = sin x + (y + 1)minus1 defined on the set [0infin)2 Although667
inf yisin[0infin) F(x y) = sin x668
and the set maxinf-points is not empty we cannot apply Ky Fan Inequality because669
the domain of F is not compact the function F(middot y) is not even sup-compact670
Detail If we consider the functions Fν(x y) = sin x + (y + 1)minus1 on the compact671
domains [0 ν]2 one can apply Ky Fanrsquos Inequality Indeed in this case we have672
inf yisin[0ν) F(x y) = sin x + (ν + 1)minus1673
that converges pointwise and hypo- to sin x and674
argmaxyisin[0ν] inf yisin[0ν] F(x y) = π2 + 2kπ
∣∣ k isin IN
675
Thus xν = π2 and xν = π2 + 2νπ are maxinf-points of the Fν The sequence676
xννisinIN converges to a maxinf-point of F the second sequence xννisinIN does not677
678
123
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References679
1 Attouch H Variational convergence for functions and operators Applicable Mathematics Series680
Pitman London (1984)681
2 Attouch H Wets RB Convergence des points minsup et de points fixes C R Acad Sci682
Paris 296 657ndash660 (1983)683
3 Attouch H Wets RB A convergence theory for saddle functions Trans Am Math Soc 280684
1ndash41 (1983)685
4 Aubin JP Ekeland I Applied nonlinear analysis Wiley London (1984)686
5 Aubin JP Frankowska H Set-valued analysis Birkhaumluser (1990)687
6 Bagh A Approximation for optimal control problems Lecture at Universidad de Chile Santiago688
(1999)689
7 Fan K A minimax inequality and applications In Shisha O (ed) InequalitiesmdashIII pp 103ndash113690
Academic Dublin (1972)691
8 Jofreacute A Rockafellar R Wets RB A variational inequality scheme for determining an economic692
equilibrium of classical or extended type In Giannessi F Maugeri A (eds) Variational analysis and693
applications pp 553ndash578 Springer New York (2005)694
9 Jofreacute A Wets RB Continuity properties of Walras equilibrium points Ann Oper Res 114 229ndash695
243 (2002)696
10 Jofreacute A Wets RB Variational convergence of bivariate functions Motivating applications II697
(Manuscript) (2006)698
11 Jofreacute A Wets RB Variational convergence of bivariate functions motivating applications I699
(Manuscript) (2004)700
12 Lignola M Morgan J Convergence of marginal functions with dependent con-701
straints Optimization 23 189ndash213 (1992)702
13 Rockafellar R Convex analysis Princeton University Press Princeton (1970)703
14 Rockafellar R Wets RB Variational analysis 2nd edn Springer New York (2004)704
123
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Example 1 [an epi-limit thats not in f v-fcn(Rn)] Consider the sequence of functions156 f ν [0infin) rarr R
νisinIN with157
f ν(x) =
⎧⎪⎨
⎪⎩
minusν2x if 0 le x le νminus1
ν2x minus 2ν if νminus1 le x le 2νminus1
0 for x ge 2νminus1
158
Detail The functions f ν isin f v-fcn(R) and for the sequence xν = νminus1 f ν(xν) rarr minusinfin159
and f ν rarre f where f [0infin) rarr R with f equiv 0 on (0infin) and f (0) = minusinfin Thus the160
functions f ν epi-converge to f as functions in fcn(R) provided they are appropriately161
extended ie taking on the value infin on (minusinfin 0) But they do not epi-converge to a162
function in f v-fcn(R) 163
In addition to the lsquogeometricrsquo definition the next proposition provides an lsquoanalyticrsquo164
characterization of epi-converging sequences in f v-fcn(Rn)165
Proposition 2 [epi-convergence in f v-fcn(Rn)] Let
f D rarr R f ν Dν rarr166
R ν isin IN
be a collection of functions in f v-fcn(Rn) Then f ν rarre f if and only the167
following conditions are satisfied168
(a) forall xν isin Dν rarr x in D liminfν f ν(xν) ge f (x)169
(ainfin) for all xν isin Dν rarr x isin D f ν(xν)infin4170
(b) forallx isin D exist xν isin Dν rarr x such that limsupν f ν(xν) le f (x)171
Proof If epi f ν rarr epi f and xν isin Dν rarr x either lim infν f ν(xν) = α lt infin or172
not Lemma 1 reminds us that α = minusinfin is not a possibility In the first instance173
(x α) is a cluster point of(xν f ν(xν)) isin epi f ν
νisinIN and thus belongs to epi f 174
ie f (x) le α and hence (a) holds α gt minusinfin since otherwise f would not be finite175
valued on D If α = infin that means that f ν(xν)infin and x cannot belong to D176
and thus (ainfin) holds On the other hand if x isin D and thus f (x) is finite there is177
a(xν αν) isin epi f ν
νisinIN such that xν isin Dν rarr x isin D and αν rarr f (x) with178
αν ge f ν(xν) ie lim supν f ν(xν) le f (x) ie (b) is also satisfied179
Conversely if (a) and (ainfin) hold and (xν αν) isin epi f ν rarr (x α) then either x isin D180
or not recall also that in view of Lemma 1 α cannot be minusinfin since we are dealing181
with epi-convergence in f v-fcn(Rn) In the latter instance by (ainfin) α = infin so we are182
not dealing with a converging sequence of points (in Rn+1) and there is no need to183
consider this situation any further When x isin D since then lim infν f ν(xν) ge f (x)184
and αν ge f ν(xν) one has α ge f (x) and consequently (x α) belongs to epi f 185
this means that condition (a-set) is satisfied If (x α) isin epi f from (b) follows the186
existence of a sequence xν isin Dν rarr x such that lim supν f ν(xν) le f (x) le α We187
can then choose the αν ge f ν(xν) so that αν rarr α that yields (b-set) the second188
condition for the set-convergence of epi f ν rarr epi f 189
Theorem 1 (epi-convergence basic properties) Consider a sequence
f ν Dν rarr190
R ν isin IN sub f v-fcn(Rn) epi-converging to f D rarr R also in f v-fcn(Rn) Then191
4 means non-decreasing and converging to ie not necessarily monotonically
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Variational convergence of bivariate functions
lim supνrarrinfin
(inf f ν) le inf f192
Moreover if xk isin argminDνk f νk for some subsequence νk and xk rarr x then193
x isin argminD f and minDνk f νk rarr minD f 194
If argminD f is a singleton then every convergent subsequence of minimizers con-195
verges to argminD f 196
Proof Let xlinfinl=1 be a sequence in D such that f (xl) rarr inf f By 2(b) for each l197
one can find a sequence xνl isin Dν rarr xl such that lim supν f ν(xνl) le f (xl) Since198
for all ν inf f ν le f ν(xνl) it follows that for all l199
lim supν
(inf f ν) le lim supν
f ν(xνl) le f (xl)200
and one has lim supν(inf f ν) le inf f since f (xl) rarr inf f 201
For the sequence xk isin Dνk rarr x from the above and 2(a)202
inf f ge lim supk
f νk (xk) ge lim infk
f νk (xk) ge f (x)203
ie x minimizes f on D and f νk (xk) = minDνk f νk rarr minD f 204
Finally since every convergent subsequence of minimizers of the functions f ν205
converges to a minimizer of f it follows that it must converge to the unique minimizer206
when argminD f is a singleton 207
In most of the applications we shall rely on a somewhat more restrictive notion208
than lsquosimplersquo epi-convergence to guarantee the convergence of the infimums209
Definition 2 (tight epi-convergence) The sequence210
f ν Dν rarr RνisinIN sub f v-fcn(Rn) epi-converges tightly to f D rarr R isin211
f v-fcn(Rn) if f ν rarre f and for all ε gt 0 there exist a compact set Bε and an in-212
dex νε such that213
forall ν ge νε inf BεcapDν f ν le inf Dν f ν + ε214
Theorem 2 (convergence of the infimums) Let f ν Dν rarr RνisinIN sub f v-fcn(Rn)215
be a sequence of functions that epi-converges to the function f D rarr R also in216
f v-fcn(Rn) with inf D f finite Then they epi-converge tightly217
(a) if and only if inf Dν f ν rarr inf D f 218
(b) if and only if there exists a sequence εν 0 such that εν-argmin f ν rarr argmin f 219
Proof Lets start with necessity in (a) For given ε gt 0 the assumptions and Theorem 1220
imply221
lim infν
( infDνcapBε
f ν) le lim infν
(infDν
f ν) + ε le lim supν
(infDν
f ν) + ε le infD
f + ε lt infin222
123
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If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223
follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224
point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225
with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226
it follows that there is no such sequences with κ arbitrarily negative In other words227
excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228
from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229
a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230
Hence231
inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232
In combination with our first string of inequalities and the fact that ε gt 0 can be233
chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234
Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235
enough given any δ gt 0 to exhibit a compact set B such that lim supν
(inf BcapDν f ν
) le236
inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237
f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238
Let B be any compact set large enough to contain all the points xν Then inf B f ν le239
f ν(xν) for all ν so B has the desired property240
We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241
tion and consequently for ν large enough also αν is finite Since convergence of242
the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243
find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244
εν = αν minus αν 245
For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246
f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247
because f ν rarre f one obtains248
inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249
le lim infν(inf f ν) le limsupν(inf f ν) le inf f250
where the last inequality comes from Theorem 1 251
Remark 1 (convergence of domains) Although epi-convergence essentially implies252
convergence of the level sets [14 Proposition 77] it does not follow that it implies253
the convergence of their (effective) domains Indeed consider the following sequence254
f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255
function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256
This vigorously argues against the temptation of involving the convergence of their257
domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258
This concludes the presentation of the results that will be used in the sequel As259
indicated earlier its also possible to derive these results from those for extended real-260
valued functions To do so one identifies f v-fcn(Rn) with261
pr -fcn(Rn) = f isin fcn(Rn)
∣∣ minus infin lt f equiv infin
262
123
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Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
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A Jofreacute R J -B Wets
η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
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Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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unco
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A Jofreacute R J -B Wets
For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
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Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
the functions gν hypo-converge to g Maxinf-points for Fν and F are then maximizers599
of the corresponding functions gν and g The assertions now follow immediately600
from the convergence of the argmax of hypo-converging sequences cf Theorem [14601
Theorem 731] translated to the lsquomaximizationrsquo framework 602
To deal with a lsquominsuprsquo situations one can either repeat all the arguments changing603
inf to sup liminf to limsup and vice-versa or simply re-integrate the questions to604
the lsquomaxinfrsquo framework by changing signs of the approximating and limit bivariate605
functions606
123
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Variational convergence of bivariate functions
4 Ky Fanrsquos Inequality extended607
The class of usc functions is closed under hypo-converge [14 Theorem 74] and so608
is the class of concave usc functions [14 Theorem 717] A class of functions that609
is closed under lopsided convergence is the class of Ky Fan functions (Theorem 8)610
We exploit this result to obtain a generalization of Ky Fan Inequality that allows us611
to claim existence of maxinf-points in situations when the domain of definition of the612
Ky Fan function is not necessarily compact613
Definition 7 A bivariate function F C times D rarr R with Cand D convex sets in614
f v-biv(Rn+m) is called a Ky Fan function if615
(a) forall y isin D x rarr F(x y) is usc on C 616
(b) forall x isin C y rarr F(x y) is convex on D617
Note that the sets C or D are not required to be compact618
Theorem 8 (lop-limits of Ky Fan functions) The lopsided limit F C times D rarr R of a619
sequence
Fν Cν times Dν rarr RνisinIN of Ky Fan functions in f v-biv(Rn+m) is also a620
Ky Fan function621
Proof For the convexity of y rarr F(x y) let xν isin Cν rarr x isin C be the sequence set622
forth by 3(b) and y0 yλ y1 isin D with yλ = (1 minus λ)y0 + λy1 for λ isin [0 1] In view623
of 3(a) we can choose sequences
y0ν isin Dν rarr y0
y1ν isin Dν rarr y1
such that624
Fν(xν y0ν) rarr F(x y0) and Fν(xν y1ν) rarr F(x y1) Let yλν = (1 minus λ)y0ν +625
λy1ν yλν isin Dν since the functions Fν(x middot) are convex and the sequence yλννisinIN626
certainly converges to yλ For all ν one has627
Fν(xν yλν) le (1 minus λ)Fν(xν y0ν) + λFν(xν y1ν)628
Taking lininf on both sides yields629
F(x yλ) le liminfν Fν(xν yλν) le (1 minus λ)F(x y0) + λF(x y1)630
that establishes the convexity of F(x middot)631
To prove the upper semicontinuity of F with respect to x-variable we show that632
for y isin D633
hypo F(middot y) is the inner set-limit of the hypo Fν(middot yν)634
where the limit is with respect to all sequences yν isin DννisinIN converging to y and635
ν rarr infin This yields the upper semicontinuity since the inner set-limit is always636
closed and a function is usc if and only if its hypograph is closed We have to show637
that if (x α) isin hypo F(middot y) then whenever yν isin Dν rarr y one can find (xν αν) isin638
hypo Fν(middot yν) such that (xν αν) rarr (x α) But that follows immediately from 3(a)639
since we can adjust the αν le Fν(xν yν) so that they converge to α le F(x y) 640
Given a Ky Fan function with compact domain and non-negative on the diagonal641
we have the following important existence result642
123
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Lemma 2 (Ky Fanrsquos Inequality [7] [4 Theorem 635]) Suppose F C times C rarr R is643
a Ky Fan function with C compact Then the set of maxinf-points of F is a nonempty644
subset of C Moreover if F(x x) ge 0 (on C times C) then for every maxinf-point x of645
C F(x middot) ge 0 on C646
One of the consequences of the lopsided convergence is an extension of the Ky Fanrsquos647
Inequality to the case when it is not possible to apply it directly because one of the648
conditions is not satisfied for example the compactness of the domain However we are649
able to approach the bivariate function F by a sequence FννisinIN defined on compact650
sets Cν This procedure could be useful in many situation where the original maxinf-651
problem is unbounded and then the problem is approached by a family of truncated652
maxinf-problems Such is the case for example when we consider as variables in653
the original problem the multipliers associated to inequality constraints or when the654
original problem is a Walras equilibrium with a positive orthant as consumption set655
in [8] one is precisely confronted with such situations Another simple illustrative656
example follows the statement of the theorem657
Theorem 9 (Extension of Ky Fanrsquos Inequality) Let empty = C sub Rn and F a finite-valued658
bivariate function defined on CtimesC Suppose one can find sequences of compact convex659
setsCν sub R
n
and (finite-valued) Ky Fan functions Fν Cν times Cν rarr RνisinIN lop-660
converging y-tightly to F then every cluster point x of any sequence xν ν isin IN of661
maxinf-points of the Fν is a maxinf-point of F662
Proof Ky Fanrsquos Inequality 2 implies that for all ν the set of maxinf-points of Fν is663
non-empty On the other hand in view of Theorems 8 and 4 any cluster point of such664
maxinf-points will be a maxinf-point of F 665
Example 2 (Extended Ky Fanrsquos Inequality applied) We consider a Ky Fan function666
F(x y) = sin x + (y + 1)minus1 defined on the set [0infin)2 Although667
inf yisin[0infin) F(x y) = sin x668
and the set maxinf-points is not empty we cannot apply Ky Fan Inequality because669
the domain of F is not compact the function F(middot y) is not even sup-compact670
Detail If we consider the functions Fν(x y) = sin x + (y + 1)minus1 on the compact671
domains [0 ν]2 one can apply Ky Fanrsquos Inequality Indeed in this case we have672
inf yisin[0ν) F(x y) = sin x + (ν + 1)minus1673
that converges pointwise and hypo- to sin x and674
argmaxyisin[0ν] inf yisin[0ν] F(x y) = π2 + 2kπ
∣∣ k isin IN
675
Thus xν = π2 and xν = π2 + 2νπ are maxinf-points of the Fν The sequence676
xννisinIN converges to a maxinf-point of F the second sequence xννisinIN does not677
678
123
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Variational convergence of bivariate functions
References679
1 Attouch H Variational convergence for functions and operators Applicable Mathematics Series680
Pitman London (1984)681
2 Attouch H Wets RB Convergence des points minsup et de points fixes C R Acad Sci682
Paris 296 657ndash660 (1983)683
3 Attouch H Wets RB A convergence theory for saddle functions Trans Am Math Soc 280684
1ndash41 (1983)685
4 Aubin JP Ekeland I Applied nonlinear analysis Wiley London (1984)686
5 Aubin JP Frankowska H Set-valued analysis Birkhaumluser (1990)687
6 Bagh A Approximation for optimal control problems Lecture at Universidad de Chile Santiago688
(1999)689
7 Fan K A minimax inequality and applications In Shisha O (ed) InequalitiesmdashIII pp 103ndash113690
Academic Dublin (1972)691
8 Jofreacute A Rockafellar R Wets RB A variational inequality scheme for determining an economic692
equilibrium of classical or extended type In Giannessi F Maugeri A (eds) Variational analysis and693
applications pp 553ndash578 Springer New York (2005)694
9 Jofreacute A Wets RB Continuity properties of Walras equilibrium points Ann Oper Res 114 229ndash695
243 (2002)696
10 Jofreacute A Wets RB Variational convergence of bivariate functions Motivating applications II697
(Manuscript) (2006)698
11 Jofreacute A Wets RB Variational convergence of bivariate functions motivating applications I699
(Manuscript) (2004)700
12 Lignola M Morgan J Convergence of marginal functions with dependent con-701
straints Optimization 23 189ndash213 (1992)702
13 Rockafellar R Convex analysis Princeton University Press Princeton (1970)703
14 Rockafellar R Wets RB Variational analysis 2nd edn Springer New York (2004)704
123
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Variational convergence of bivariate functions
lim supνrarrinfin
(inf f ν) le inf f192
Moreover if xk isin argminDνk f νk for some subsequence νk and xk rarr x then193
x isin argminD f and minDνk f νk rarr minD f 194
If argminD f is a singleton then every convergent subsequence of minimizers con-195
verges to argminD f 196
Proof Let xlinfinl=1 be a sequence in D such that f (xl) rarr inf f By 2(b) for each l197
one can find a sequence xνl isin Dν rarr xl such that lim supν f ν(xνl) le f (xl) Since198
for all ν inf f ν le f ν(xνl) it follows that for all l199
lim supν
(inf f ν) le lim supν
f ν(xνl) le f (xl)200
and one has lim supν(inf f ν) le inf f since f (xl) rarr inf f 201
For the sequence xk isin Dνk rarr x from the above and 2(a)202
inf f ge lim supk
f νk (xk) ge lim infk
f νk (xk) ge f (x)203
ie x minimizes f on D and f νk (xk) = minDνk f νk rarr minD f 204
Finally since every convergent subsequence of minimizers of the functions f ν205
converges to a minimizer of f it follows that it must converge to the unique minimizer206
when argminD f is a singleton 207
In most of the applications we shall rely on a somewhat more restrictive notion208
than lsquosimplersquo epi-convergence to guarantee the convergence of the infimums209
Definition 2 (tight epi-convergence) The sequence210
f ν Dν rarr RνisinIN sub f v-fcn(Rn) epi-converges tightly to f D rarr R isin211
f v-fcn(Rn) if f ν rarre f and for all ε gt 0 there exist a compact set Bε and an in-212
dex νε such that213
forall ν ge νε inf BεcapDν f ν le inf Dν f ν + ε214
Theorem 2 (convergence of the infimums) Let f ν Dν rarr RνisinIN sub f v-fcn(Rn)215
be a sequence of functions that epi-converges to the function f D rarr R also in216
f v-fcn(Rn) with inf D f finite Then they epi-converge tightly217
(a) if and only if inf Dν f ν rarr inf D f 218
(b) if and only if there exists a sequence εν 0 such that εν-argmin f ν rarr argmin f 219
Proof Lets start with necessity in (a) For given ε gt 0 the assumptions and Theorem 1220
imply221
lim infν
( infDνcapBε
f ν) le lim infν
(infDν
f ν) + ε le lim supν
(infDν
f ν) + ε le infD
f + ε lt infin222
123
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A Jofreacute R J -B Wets
If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223
follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224
point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225
with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226
it follows that there is no such sequences with κ arbitrarily negative In other words227
excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228
from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229
a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230
Hence231
inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232
In combination with our first string of inequalities and the fact that ε gt 0 can be233
chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234
Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235
enough given any δ gt 0 to exhibit a compact set B such that lim supν
(inf BcapDν f ν
) le236
inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237
f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238
Let B be any compact set large enough to contain all the points xν Then inf B f ν le239
f ν(xν) for all ν so B has the desired property240
We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241
tion and consequently for ν large enough also αν is finite Since convergence of242
the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243
find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244
εν = αν minus αν 245
For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246
f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247
because f ν rarre f one obtains248
inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249
le lim infν(inf f ν) le limsupν(inf f ν) le inf f250
where the last inequality comes from Theorem 1 251
Remark 1 (convergence of domains) Although epi-convergence essentially implies252
convergence of the level sets [14 Proposition 77] it does not follow that it implies253
the convergence of their (effective) domains Indeed consider the following sequence254
f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255
function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256
This vigorously argues against the temptation of involving the convergence of their257
domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258
This concludes the presentation of the results that will be used in the sequel As259
indicated earlier its also possible to derive these results from those for extended real-260
valued functions To do so one identifies f v-fcn(Rn) with261
pr -fcn(Rn) = f isin fcn(Rn)
∣∣ minus infin lt f equiv infin
262
123
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Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
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η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
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Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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unco
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For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
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Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
the functions gν hypo-converge to g Maxinf-points for Fν and F are then maximizers599
of the corresponding functions gν and g The assertions now follow immediately600
from the convergence of the argmax of hypo-converging sequences cf Theorem [14601
Theorem 731] translated to the lsquomaximizationrsquo framework 602
To deal with a lsquominsuprsquo situations one can either repeat all the arguments changing603
inf to sup liminf to limsup and vice-versa or simply re-integrate the questions to604
the lsquomaxinfrsquo framework by changing signs of the approximating and limit bivariate605
functions606
123
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4 Ky Fanrsquos Inequality extended607
The class of usc functions is closed under hypo-converge [14 Theorem 74] and so608
is the class of concave usc functions [14 Theorem 717] A class of functions that609
is closed under lopsided convergence is the class of Ky Fan functions (Theorem 8)610
We exploit this result to obtain a generalization of Ky Fan Inequality that allows us611
to claim existence of maxinf-points in situations when the domain of definition of the612
Ky Fan function is not necessarily compact613
Definition 7 A bivariate function F C times D rarr R with Cand D convex sets in614
f v-biv(Rn+m) is called a Ky Fan function if615
(a) forall y isin D x rarr F(x y) is usc on C 616
(b) forall x isin C y rarr F(x y) is convex on D617
Note that the sets C or D are not required to be compact618
Theorem 8 (lop-limits of Ky Fan functions) The lopsided limit F C times D rarr R of a619
sequence
Fν Cν times Dν rarr RνisinIN of Ky Fan functions in f v-biv(Rn+m) is also a620
Ky Fan function621
Proof For the convexity of y rarr F(x y) let xν isin Cν rarr x isin C be the sequence set622
forth by 3(b) and y0 yλ y1 isin D with yλ = (1 minus λ)y0 + λy1 for λ isin [0 1] In view623
of 3(a) we can choose sequences
y0ν isin Dν rarr y0
y1ν isin Dν rarr y1
such that624
Fν(xν y0ν) rarr F(x y0) and Fν(xν y1ν) rarr F(x y1) Let yλν = (1 minus λ)y0ν +625
λy1ν yλν isin Dν since the functions Fν(x middot) are convex and the sequence yλννisinIN626
certainly converges to yλ For all ν one has627
Fν(xν yλν) le (1 minus λ)Fν(xν y0ν) + λFν(xν y1ν)628
Taking lininf on both sides yields629
F(x yλ) le liminfν Fν(xν yλν) le (1 minus λ)F(x y0) + λF(x y1)630
that establishes the convexity of F(x middot)631
To prove the upper semicontinuity of F with respect to x-variable we show that632
for y isin D633
hypo F(middot y) is the inner set-limit of the hypo Fν(middot yν)634
where the limit is with respect to all sequences yν isin DννisinIN converging to y and635
ν rarr infin This yields the upper semicontinuity since the inner set-limit is always636
closed and a function is usc if and only if its hypograph is closed We have to show637
that if (x α) isin hypo F(middot y) then whenever yν isin Dν rarr y one can find (xν αν) isin638
hypo Fν(middot yν) such that (xν αν) rarr (x α) But that follows immediately from 3(a)639
since we can adjust the αν le Fν(xν yν) so that they converge to α le F(x y) 640
Given a Ky Fan function with compact domain and non-negative on the diagonal641
we have the following important existence result642
123
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Lemma 2 (Ky Fanrsquos Inequality [7] [4 Theorem 635]) Suppose F C times C rarr R is643
a Ky Fan function with C compact Then the set of maxinf-points of F is a nonempty644
subset of C Moreover if F(x x) ge 0 (on C times C) then for every maxinf-point x of645
C F(x middot) ge 0 on C646
One of the consequences of the lopsided convergence is an extension of the Ky Fanrsquos647
Inequality to the case when it is not possible to apply it directly because one of the648
conditions is not satisfied for example the compactness of the domain However we are649
able to approach the bivariate function F by a sequence FννisinIN defined on compact650
sets Cν This procedure could be useful in many situation where the original maxinf-651
problem is unbounded and then the problem is approached by a family of truncated652
maxinf-problems Such is the case for example when we consider as variables in653
the original problem the multipliers associated to inequality constraints or when the654
original problem is a Walras equilibrium with a positive orthant as consumption set655
in [8] one is precisely confronted with such situations Another simple illustrative656
example follows the statement of the theorem657
Theorem 9 (Extension of Ky Fanrsquos Inequality) Let empty = C sub Rn and F a finite-valued658
bivariate function defined on CtimesC Suppose one can find sequences of compact convex659
setsCν sub R
n
and (finite-valued) Ky Fan functions Fν Cν times Cν rarr RνisinIN lop-660
converging y-tightly to F then every cluster point x of any sequence xν ν isin IN of661
maxinf-points of the Fν is a maxinf-point of F662
Proof Ky Fanrsquos Inequality 2 implies that for all ν the set of maxinf-points of Fν is663
non-empty On the other hand in view of Theorems 8 and 4 any cluster point of such664
maxinf-points will be a maxinf-point of F 665
Example 2 (Extended Ky Fanrsquos Inequality applied) We consider a Ky Fan function666
F(x y) = sin x + (y + 1)minus1 defined on the set [0infin)2 Although667
inf yisin[0infin) F(x y) = sin x668
and the set maxinf-points is not empty we cannot apply Ky Fan Inequality because669
the domain of F is not compact the function F(middot y) is not even sup-compact670
Detail If we consider the functions Fν(x y) = sin x + (y + 1)minus1 on the compact671
domains [0 ν]2 one can apply Ky Fanrsquos Inequality Indeed in this case we have672
inf yisin[0ν) F(x y) = sin x + (ν + 1)minus1673
that converges pointwise and hypo- to sin x and674
argmaxyisin[0ν] inf yisin[0ν] F(x y) = π2 + 2kπ
∣∣ k isin IN
675
Thus xν = π2 and xν = π2 + 2νπ are maxinf-points of the Fν The sequence676
xννisinIN converges to a maxinf-point of F the second sequence xννisinIN does not677
678
123
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References679
1 Attouch H Variational convergence for functions and operators Applicable Mathematics Series680
Pitman London (1984)681
2 Attouch H Wets RB Convergence des points minsup et de points fixes C R Acad Sci682
Paris 296 657ndash660 (1983)683
3 Attouch H Wets RB A convergence theory for saddle functions Trans Am Math Soc 280684
1ndash41 (1983)685
4 Aubin JP Ekeland I Applied nonlinear analysis Wiley London (1984)686
5 Aubin JP Frankowska H Set-valued analysis Birkhaumluser (1990)687
6 Bagh A Approximation for optimal control problems Lecture at Universidad de Chile Santiago688
(1999)689
7 Fan K A minimax inequality and applications In Shisha O (ed) InequalitiesmdashIII pp 103ndash113690
Academic Dublin (1972)691
8 Jofreacute A Rockafellar R Wets RB A variational inequality scheme for determining an economic692
equilibrium of classical or extended type In Giannessi F Maugeri A (eds) Variational analysis and693
applications pp 553ndash578 Springer New York (2005)694
9 Jofreacute A Wets RB Continuity properties of Walras equilibrium points Ann Oper Res 114 229ndash695
243 (2002)696
10 Jofreacute A Wets RB Variational convergence of bivariate functions Motivating applications II697
(Manuscript) (2006)698
11 Jofreacute A Wets RB Variational convergence of bivariate functions motivating applications I699
(Manuscript) (2004)700
12 Lignola M Morgan J Convergence of marginal functions with dependent con-701
straints Optimization 23 189ndash213 (1992)702
13 Rockafellar R Convex analysis Princeton University Press Princeton (1970)703
14 Rockafellar R Wets RB Variational analysis 2nd edn Springer New York (2004)704
123
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If there is a subsequence νk such that f (xk) lt κ for some xk isin Dνk cap Bε it would223
follow that inf D f lt κ Indeed since Bε is compact the sequence xk has a cluster224
point say x and then conditions (ainfin) and (a) of Proposition 2 guarantee f (x) lt κ225
with x isin D and consequently also inf D f lt κ Since its assumed that inf D f is finite226
it follows that there is no such sequences with κ arbitrarily negative In other words227
excluding possibly a finite number of indexes the inf DνcapBε f ν stay bounded away228
from minusinfin and one can find xν isin ε- argminDνcapBεf ν The sequence xννisinIN admits229
a cluster point say x that lies in Bε and again by 2(ainfina) f (x) le liminfν f ν(xν)230
Hence231
inf D f minus ε le f (x) minus ε le liminfν f ν(xν) minus ε le liminfν(inf Dν f ν)232
In combination with our first string of inequalities and the fact that ε gt 0 can be233
chosen arbitrarily small it follows that indeed inf Dν f ν rarr inf D f 234
Next we turn to sufficiency in (a) Since inf f ν rarr inf f isin R by assumption its235
enough given any δ gt 0 to exhibit a compact set B such that lim supν
(inf BcapDν f ν
) le236
inf D f + δ Choose any point x such that f (x) le inf D f + δ Because f ν rarre f in237
f v-fcn(Rn) there exists a sequence 2(a) xν rarr x such that lim supν f ν(xν) le f (x)238
Let B be any compact set large enough to contain all the points xν Then inf B f ν le239
f ν(xν) for all ν so B has the desired property240
We derive (b) from (a) Let αν = inf f ν rarr inf f = α that is finite by assump-241
tion and consequently for ν large enough also αν is finite Since convergence of242
the epigraphs implies the convergence of the level sets [14 Proposition 77] one can243
find a sequence of αν α such that levαν f ν rarr levα f = argmin f Simply set244
εν = αν minus αν 245
For the converse suppose there is a sequence εν 0 with εν-argmin f ν rarr argmin246
f = empty For any x isin argmin f one can select xν isin εν-argmin f ν with xν rarr x Then247
because f ν rarre f one obtains248
inf f = f (x) le lim infν f ν(xν) le lim infν(inf f ν + εν)249
le lim infν(inf f ν) le limsupν(inf f ν) le inf f250
where the last inequality comes from Theorem 1 251
Remark 1 (convergence of domains) Although epi-convergence essentially implies252
convergence of the level sets [14 Proposition 77] it does not follow that it implies253
the convergence of their (effective) domains Indeed consider the following sequence254
f ν R rarr R with f ν equiv ν except for f ν(0) = 0 that epi-converges to δ0 the indicator255
function of 0 We definitely do not have dom f ν = R converging to dom δ0 = 0256
This vigorously argues against the temptation of involving the convergence of their257
domains in the definition of epi-convergence even for functions in f v-fcn(Rn)258
This concludes the presentation of the results that will be used in the sequel As259
indicated earlier its also possible to derive these results from those for extended real-260
valued functions To do so one identifies f v-fcn(Rn) with261
pr -fcn(Rn) = f isin fcn(Rn)
∣∣ minus infin lt f equiv infin
262
123
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Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
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η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
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Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
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Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
the functions gν hypo-converge to g Maxinf-points for Fν and F are then maximizers599
of the corresponding functions gν and g The assertions now follow immediately600
from the convergence of the argmax of hypo-converging sequences cf Theorem [14601
Theorem 731] translated to the lsquomaximizationrsquo framework 602
To deal with a lsquominsuprsquo situations one can either repeat all the arguments changing603
inf to sup liminf to limsup and vice-versa or simply re-integrate the questions to604
the lsquomaxinfrsquo framework by changing signs of the approximating and limit bivariate605
functions606
123
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unco
rrec
ted
proo
f
Variational convergence of bivariate functions
4 Ky Fanrsquos Inequality extended607
The class of usc functions is closed under hypo-converge [14 Theorem 74] and so608
is the class of concave usc functions [14 Theorem 717] A class of functions that609
is closed under lopsided convergence is the class of Ky Fan functions (Theorem 8)610
We exploit this result to obtain a generalization of Ky Fan Inequality that allows us611
to claim existence of maxinf-points in situations when the domain of definition of the612
Ky Fan function is not necessarily compact613
Definition 7 A bivariate function F C times D rarr R with Cand D convex sets in614
f v-biv(Rn+m) is called a Ky Fan function if615
(a) forall y isin D x rarr F(x y) is usc on C 616
(b) forall x isin C y rarr F(x y) is convex on D617
Note that the sets C or D are not required to be compact618
Theorem 8 (lop-limits of Ky Fan functions) The lopsided limit F C times D rarr R of a619
sequence
Fν Cν times Dν rarr RνisinIN of Ky Fan functions in f v-biv(Rn+m) is also a620
Ky Fan function621
Proof For the convexity of y rarr F(x y) let xν isin Cν rarr x isin C be the sequence set622
forth by 3(b) and y0 yλ y1 isin D with yλ = (1 minus λ)y0 + λy1 for λ isin [0 1] In view623
of 3(a) we can choose sequences
y0ν isin Dν rarr y0
y1ν isin Dν rarr y1
such that624
Fν(xν y0ν) rarr F(x y0) and Fν(xν y1ν) rarr F(x y1) Let yλν = (1 minus λ)y0ν +625
λy1ν yλν isin Dν since the functions Fν(x middot) are convex and the sequence yλννisinIN626
certainly converges to yλ For all ν one has627
Fν(xν yλν) le (1 minus λ)Fν(xν y0ν) + λFν(xν y1ν)628
Taking lininf on both sides yields629
F(x yλ) le liminfν Fν(xν yλν) le (1 minus λ)F(x y0) + λF(x y1)630
that establishes the convexity of F(x middot)631
To prove the upper semicontinuity of F with respect to x-variable we show that632
for y isin D633
hypo F(middot y) is the inner set-limit of the hypo Fν(middot yν)634
where the limit is with respect to all sequences yν isin DννisinIN converging to y and635
ν rarr infin This yields the upper semicontinuity since the inner set-limit is always636
closed and a function is usc if and only if its hypograph is closed We have to show637
that if (x α) isin hypo F(middot y) then whenever yν isin Dν rarr y one can find (xν αν) isin638
hypo Fν(middot yν) such that (xν αν) rarr (x α) But that follows immediately from 3(a)639
since we can adjust the αν le Fν(xν yν) so that they converge to α le F(x y) 640
Given a Ky Fan function with compact domain and non-negative on the diagonal641
we have the following important existence result642
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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Lemma 2 (Ky Fanrsquos Inequality [7] [4 Theorem 635]) Suppose F C times C rarr R is643
a Ky Fan function with C compact Then the set of maxinf-points of F is a nonempty644
subset of C Moreover if F(x x) ge 0 (on C times C) then for every maxinf-point x of645
C F(x middot) ge 0 on C646
One of the consequences of the lopsided convergence is an extension of the Ky Fanrsquos647
Inequality to the case when it is not possible to apply it directly because one of the648
conditions is not satisfied for example the compactness of the domain However we are649
able to approach the bivariate function F by a sequence FννisinIN defined on compact650
sets Cν This procedure could be useful in many situation where the original maxinf-651
problem is unbounded and then the problem is approached by a family of truncated652
maxinf-problems Such is the case for example when we consider as variables in653
the original problem the multipliers associated to inequality constraints or when the654
original problem is a Walras equilibrium with a positive orthant as consumption set655
in [8] one is precisely confronted with such situations Another simple illustrative656
example follows the statement of the theorem657
Theorem 9 (Extension of Ky Fanrsquos Inequality) Let empty = C sub Rn and F a finite-valued658
bivariate function defined on CtimesC Suppose one can find sequences of compact convex659
setsCν sub R
n
and (finite-valued) Ky Fan functions Fν Cν times Cν rarr RνisinIN lop-660
converging y-tightly to F then every cluster point x of any sequence xν ν isin IN of661
maxinf-points of the Fν is a maxinf-point of F662
Proof Ky Fanrsquos Inequality 2 implies that for all ν the set of maxinf-points of Fν is663
non-empty On the other hand in view of Theorems 8 and 4 any cluster point of such664
maxinf-points will be a maxinf-point of F 665
Example 2 (Extended Ky Fanrsquos Inequality applied) We consider a Ky Fan function666
F(x y) = sin x + (y + 1)minus1 defined on the set [0infin)2 Although667
inf yisin[0infin) F(x y) = sin x668
and the set maxinf-points is not empty we cannot apply Ky Fan Inequality because669
the domain of F is not compact the function F(middot y) is not even sup-compact670
Detail If we consider the functions Fν(x y) = sin x + (y + 1)minus1 on the compact671
domains [0 ν]2 one can apply Ky Fanrsquos Inequality Indeed in this case we have672
inf yisin[0ν) F(x y) = sin x + (ν + 1)minus1673
that converges pointwise and hypo- to sin x and674
argmaxyisin[0ν] inf yisin[0ν] F(x y) = π2 + 2kπ
∣∣ k isin IN
675
Thus xν = π2 and xν = π2 + 2νπ are maxinf-points of the Fν The sequence676
xννisinIN converges to a maxinf-point of F the second sequence xννisinIN does not677
678
123
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unco
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ted
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f
Variational convergence of bivariate functions
References679
1 Attouch H Variational convergence for functions and operators Applicable Mathematics Series680
Pitman London (1984)681
2 Attouch H Wets RB Convergence des points minsup et de points fixes C R Acad Sci682
Paris 296 657ndash660 (1983)683
3 Attouch H Wets RB A convergence theory for saddle functions Trans Am Math Soc 280684
1ndash41 (1983)685
4 Aubin JP Ekeland I Applied nonlinear analysis Wiley London (1984)686
5 Aubin JP Frankowska H Set-valued analysis Birkhaumluser (1990)687
6 Bagh A Approximation for optimal control problems Lecture at Universidad de Chile Santiago688
(1999)689
7 Fan K A minimax inequality and applications In Shisha O (ed) InequalitiesmdashIII pp 103ndash113690
Academic Dublin (1972)691
8 Jofreacute A Rockafellar R Wets RB A variational inequality scheme for determining an economic692
equilibrium of classical or extended type In Giannessi F Maugeri A (eds) Variational analysis and693
applications pp 553ndash578 Springer New York (2005)694
9 Jofreacute A Wets RB Continuity properties of Walras equilibrium points Ann Oper Res 114 229ndash695
243 (2002)696
10 Jofreacute A Wets RB Variational convergence of bivariate functions Motivating applications II697
(Manuscript) (2006)698
11 Jofreacute A Wets RB Variational convergence of bivariate functions motivating applications I699
(Manuscript) (2004)700
12 Lignola M Morgan J Convergence of marginal functions with dependent con-701
straints Optimization 23 189ndash213 (1992)702
13 Rockafellar R Convex analysis Princeton University Press Princeton (1970)703
14 Rockafellar R Wets RB Variational analysis 2nd edn Springer New York (2004)704
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
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f
Variational convergence of bivariate functions
the subset of proper functions in fcn(Rn) in a minimization context a function f is263
said to be proper if f gt minusinfin and f equiv infin in which case its finite on its (effective)264
domain265
dom f = x isin R
n∣∣ f (x) lt infin
266
There is an one-to-one correspondence a bijection5 denoted η between the elements267
of f v-fcn(Rn) and those of pr -fcn(Rn) If f isin f v-fcn(Rn) its extension to all of268
Rn by setting η f = f on its domain and η f equiv infin on the complement of its domain269
uniquely identifies a function in pr -fcn(Rn) And if f isin pr -fcn(Rn) the restriction270
of f to dom f uniquely identifies a function ηminus1 f in f v-fcn(Rn) Its important to271
observe that under this bijection any function either in pr -fcn(Rn) or f v-fcn(Rn)272
and the corresponding one in f v-fcn(Rn) or pr -fcn(Rn) have the same epigraphs273
Since epi-convergence for sequences in f v-fcn(Rn) or in in fcn(Rn) is always274
defined in terms of the convergence of the epigraphs there is really no need to verify275
that the analytic versions (Proposition 2 and [14 Proposition 72]) also coincide276
However for completeness sake and to highlight the connections we go through the277
details of an argument278
Proposition 3 (epi-convergence in f v-fcn(Rn) and fcn(Rn)) Let279 f D rarr R f ν Dν rarr R ν isin IN
be a collection of functions in f v-fcn(Rn)280
Then f ν rarre f if and only η f ν rarre η f where η is the bijection defined above281
Proof Now η f ν rarre η f ([14 Proposition 72]) if and only if for all x isin Rn 282
(aη) liminfν η f ν(xν) ge η f (x) for every sequence xν rarr x 283
(bη) limsupν η f ν(xν) le η f (x) for some sequence xν rarr x 284
Since for x isin D η f (x) = infin (aη) clearly implies (a) amp (ainfin) Conversely if (a) and285
(ainfin) hold x isin D and xν rarr x when computing the liminfν η f ν(xν) one can ignore286
elements xν isin Dν since then η f ν(xν) = infin Hence for x isin D actually (a) implies287
(aη) If x isin D and xν rarr x (ainfin) and again the fact that η f ν(xν) = infin when x isin Dν 288
yield (aη)289
If (bη) hold and x isin D then the sequence xν rarr x must at least eventually have290
xν isin Dν since otherwise the limsupν η f ν(xν) would be infin whereas f (x) = η f (x) is291
finite Thus (bη) implies (b) Conversely (b) certainly yields (bη) if x isin D If x isin D292
η f (x) = infin and so the inequality in (bη) is also trivially satisfied in that case 293
As long as we restrict our attention to pr -fcn(Rn) in view of the preceding294
observations all the basic results cf [14 Chap 7 Sect E] of the theory of epi-295
convergence related to the convergence of infimums and minimizers apply equally296
well to functions in f v-fcn(Rn) and not just those featured here In particular if one297
takes into account the bijection between f v-fcn(Rn) and pr -fcn(Rn) then Theorem 1298
is simply an adaptation of the standard results for epi-converging sequences in fcn(Rn)299
cf [14 Proposition 730 Theorem 731] Similarly again by relying on the bijection300
5 In fact this bijection is a homeomorphism when we restrict our attention to lsc functions The continuityof this correspondence is immediate if both of these function-spaces are equipped with the topology inducedby the convergence of the epigraphs see below
123
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ted
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f
A Jofreacute R J -B Wets
η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
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Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
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unco
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f
Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
the functions gν hypo-converge to g Maxinf-points for Fν and F are then maximizers599
of the corresponding functions gν and g The assertions now follow immediately600
from the convergence of the argmax of hypo-converging sequences cf Theorem [14601
Theorem 731] translated to the lsquomaximizationrsquo framework 602
To deal with a lsquominsuprsquo situations one can either repeat all the arguments changing603
inf to sup liminf to limsup and vice-versa or simply re-integrate the questions to604
the lsquomaxinfrsquo framework by changing signs of the approximating and limit bivariate605
functions606
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
rrec
ted
proo
f
Variational convergence of bivariate functions
4 Ky Fanrsquos Inequality extended607
The class of usc functions is closed under hypo-converge [14 Theorem 74] and so608
is the class of concave usc functions [14 Theorem 717] A class of functions that609
is closed under lopsided convergence is the class of Ky Fan functions (Theorem 8)610
We exploit this result to obtain a generalization of Ky Fan Inequality that allows us611
to claim existence of maxinf-points in situations when the domain of definition of the612
Ky Fan function is not necessarily compact613
Definition 7 A bivariate function F C times D rarr R with Cand D convex sets in614
f v-biv(Rn+m) is called a Ky Fan function if615
(a) forall y isin D x rarr F(x y) is usc on C 616
(b) forall x isin C y rarr F(x y) is convex on D617
Note that the sets C or D are not required to be compact618
Theorem 8 (lop-limits of Ky Fan functions) The lopsided limit F C times D rarr R of a619
sequence
Fν Cν times Dν rarr RνisinIN of Ky Fan functions in f v-biv(Rn+m) is also a620
Ky Fan function621
Proof For the convexity of y rarr F(x y) let xν isin Cν rarr x isin C be the sequence set622
forth by 3(b) and y0 yλ y1 isin D with yλ = (1 minus λ)y0 + λy1 for λ isin [0 1] In view623
of 3(a) we can choose sequences
y0ν isin Dν rarr y0
y1ν isin Dν rarr y1
such that624
Fν(xν y0ν) rarr F(x y0) and Fν(xν y1ν) rarr F(x y1) Let yλν = (1 minus λ)y0ν +625
λy1ν yλν isin Dν since the functions Fν(x middot) are convex and the sequence yλννisinIN626
certainly converges to yλ For all ν one has627
Fν(xν yλν) le (1 minus λ)Fν(xν y0ν) + λFν(xν y1ν)628
Taking lininf on both sides yields629
F(x yλ) le liminfν Fν(xν yλν) le (1 minus λ)F(x y0) + λF(x y1)630
that establishes the convexity of F(x middot)631
To prove the upper semicontinuity of F with respect to x-variable we show that632
for y isin D633
hypo F(middot y) is the inner set-limit of the hypo Fν(middot yν)634
where the limit is with respect to all sequences yν isin DννisinIN converging to y and635
ν rarr infin This yields the upper semicontinuity since the inner set-limit is always636
closed and a function is usc if and only if its hypograph is closed We have to show637
that if (x α) isin hypo F(middot y) then whenever yν isin Dν rarr y one can find (xν αν) isin638
hypo Fν(middot yν) such that (xν αν) rarr (x α) But that follows immediately from 3(a)639
since we can adjust the αν le Fν(xν yν) so that they converge to α le F(x y) 640
Given a Ky Fan function with compact domain and non-negative on the diagonal641
we have the following important existence result642
123
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unco
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A Jofreacute R J -B Wets
Lemma 2 (Ky Fanrsquos Inequality [7] [4 Theorem 635]) Suppose F C times C rarr R is643
a Ky Fan function with C compact Then the set of maxinf-points of F is a nonempty644
subset of C Moreover if F(x x) ge 0 (on C times C) then for every maxinf-point x of645
C F(x middot) ge 0 on C646
One of the consequences of the lopsided convergence is an extension of the Ky Fanrsquos647
Inequality to the case when it is not possible to apply it directly because one of the648
conditions is not satisfied for example the compactness of the domain However we are649
able to approach the bivariate function F by a sequence FννisinIN defined on compact650
sets Cν This procedure could be useful in many situation where the original maxinf-651
problem is unbounded and then the problem is approached by a family of truncated652
maxinf-problems Such is the case for example when we consider as variables in653
the original problem the multipliers associated to inequality constraints or when the654
original problem is a Walras equilibrium with a positive orthant as consumption set655
in [8] one is precisely confronted with such situations Another simple illustrative656
example follows the statement of the theorem657
Theorem 9 (Extension of Ky Fanrsquos Inequality) Let empty = C sub Rn and F a finite-valued658
bivariate function defined on CtimesC Suppose one can find sequences of compact convex659
setsCν sub R
n
and (finite-valued) Ky Fan functions Fν Cν times Cν rarr RνisinIN lop-660
converging y-tightly to F then every cluster point x of any sequence xν ν isin IN of661
maxinf-points of the Fν is a maxinf-point of F662
Proof Ky Fanrsquos Inequality 2 implies that for all ν the set of maxinf-points of Fν is663
non-empty On the other hand in view of Theorems 8 and 4 any cluster point of such664
maxinf-points will be a maxinf-point of F 665
Example 2 (Extended Ky Fanrsquos Inequality applied) We consider a Ky Fan function666
F(x y) = sin x + (y + 1)minus1 defined on the set [0infin)2 Although667
inf yisin[0infin) F(x y) = sin x668
and the set maxinf-points is not empty we cannot apply Ky Fan Inequality because669
the domain of F is not compact the function F(middot y) is not even sup-compact670
Detail If we consider the functions Fν(x y) = sin x + (y + 1)minus1 on the compact671
domains [0 ν]2 one can apply Ky Fanrsquos Inequality Indeed in this case we have672
inf yisin[0ν) F(x y) = sin x + (ν + 1)minus1673
that converges pointwise and hypo- to sin x and674
argmaxyisin[0ν] inf yisin[0ν] F(x y) = π2 + 2kπ
∣∣ k isin IN
675
Thus xν = π2 and xν = π2 + 2νπ are maxinf-points of the Fν The sequence676
xννisinIN converges to a maxinf-point of F the second sequence xννisinIN does not677
678
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
rrec
ted
proo
f
Variational convergence of bivariate functions
References679
1 Attouch H Variational convergence for functions and operators Applicable Mathematics Series680
Pitman London (1984)681
2 Attouch H Wets RB Convergence des points minsup et de points fixes C R Acad Sci682
Paris 296 657ndash660 (1983)683
3 Attouch H Wets RB A convergence theory for saddle functions Trans Am Math Soc 280684
1ndash41 (1983)685
4 Aubin JP Ekeland I Applied nonlinear analysis Wiley London (1984)686
5 Aubin JP Frankowska H Set-valued analysis Birkhaumluser (1990)687
6 Bagh A Approximation for optimal control problems Lecture at Universidad de Chile Santiago688
(1999)689
7 Fan K A minimax inequality and applications In Shisha O (ed) InequalitiesmdashIII pp 103ndash113690
Academic Dublin (1972)691
8 Jofreacute A Rockafellar R Wets RB A variational inequality scheme for determining an economic692
equilibrium of classical or extended type In Giannessi F Maugeri A (eds) Variational analysis and693
applications pp 553ndash578 Springer New York (2005)694
9 Jofreacute A Wets RB Continuity properties of Walras equilibrium points Ann Oper Res 114 229ndash695
243 (2002)696
10 Jofreacute A Wets RB Variational convergence of bivariate functions Motivating applications II697
(Manuscript) (2006)698
11 Jofreacute A Wets RB Variational convergence of bivariate functions motivating applications I699
(Manuscript) (2004)700
12 Lignola M Morgan J Convergence of marginal functions with dependent con-701
straints Optimization 23 189ndash213 (1992)702
13 Rockafellar R Convex analysis Princeton University Press Princeton (1970)703
14 Rockafellar R Wets RB Variational analysis 2nd edn Springer New York (2004)704
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
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ted
proo
f
A Jofreacute R J -B Wets
η to translate the statement of Theorem 2 into an equivalent one for functions η f ν η f301
that belong to fcn(Rn) one comes up with [14 Theorem 731] about the convergence302
of the infimal values303
Finally in a maximization setting one can simply pass from f to minus f or one can304
repeat the previous arguments with the following changes in the terminology min to305
max (inf to sup) infin to minusinfin epi to hypo le to ge (and vice-versa) lim inf to lim sup306
(and vice-versa) and lsc to usc The hypograph of f is the set of all points in Rn+1
307
that lie on or below the graph of f f is usc (=upper semicontinuous) if its hypograph308
is closed and its proper in the maximization framework if minusinfin equiv f lt infin in309
the maximization setting cl f denotes the function whose hypograph is the closure310
relative to Rn+1 of hypo f its also called its usc regularization311
A sequence is said to hypo-converge written f ν rarrh f when minus f ν rarre minus f or312
equivalently if hypo f ν rarr hypo f and it hypo-converge tightly if minus f ν epi-converge313
tightly to minus f And consequently if the sequence hypo-converges tightly to f with314
supD f finite then supDν f ν rarr supD f 315
When hypo f is the inner set-limit of the hypo f ν then f is the lower hypo-limit316
of the functions f ν and if its the outer set-limit then its their upper hypo-limit It then317
follows from Proposition 1 that the lower and upper hypo-limits and the hypo-limit318
if it exists are all usc Moreover if the functions f ν are concave so is the lower319
hypo-limit and the hypo-limit if it exists Hence one also has that the family of usc320
functions is closed under hypo-convergence321
3 Lopsided convergence322
Lopsided convergence for bivariate functions was introduced in [2] we already relied323
on this notion to formalize the convergence of pure exchange economies and to study324
the stability of their Walras equilibrium points [9] Its aimed at the convergence of325
maxinf-points or minsup-points but not at both therefore the name lopsided or lop-326
convergence However our present more comprehensive analysis has lead us to adjust327
the definition since otherwise some lsquonaturalrsquo classes of bivariate functions with domain328
and values like those depicted in Fig 2 would essentially be excluded ie could not329
be included in (lopsided or) lop-convergent families And these are precisely the class330
of functions that needs to be dealt with in many applications Moreover like in Sect331
2 the main focus will not be on extended real-valued functions but on finite-valued332
bivariate functions that are only defined on a product of non-empty sets rather than333
on extended real-valued functions defined on the full product space The motivation334
for proceeding in this manner again coming from the applications But this time its335
not just one possible approach its in fact mandated by the underlying structure of the336
class of bivariates that are of interest in the applications We shall however like in the337
previous section provide the bridge with the lsquoextended real-valuedrsquo framework that338
was used in [2]339
The definition of lop-convergence is necessarily one-sided One is either inter-340
ested in the convergence of maxinf-points or minsup-points but not both In general341
the maxinf-points are not minsup-points and vice-versa When they identify the342
same points such points are saddle-points In this article our concern is with the343
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
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ted
proo
f
Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
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ted
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f
A Jofreacute R J -B Wets
For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
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unco
rrec
ted
proo
f
Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404
the functions gν hypo-converge to g Maxinf-points for Fν and F are then maximizers599
of the corresponding functions gν and g The assertions now follow immediately600
from the convergence of the argmax of hypo-converging sequences cf Theorem [14601
Theorem 731] translated to the lsquomaximizationrsquo framework 602
To deal with a lsquominsuprsquo situations one can either repeat all the arguments changing603
inf to sup liminf to limsup and vice-versa or simply re-integrate the questions to604
the lsquomaxinfrsquo framework by changing signs of the approximating and limit bivariate605
functions606
123
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unco
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ted
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Variational convergence of bivariate functions
4 Ky Fanrsquos Inequality extended607
The class of usc functions is closed under hypo-converge [14 Theorem 74] and so608
is the class of concave usc functions [14 Theorem 717] A class of functions that609
is closed under lopsided convergence is the class of Ky Fan functions (Theorem 8)610
We exploit this result to obtain a generalization of Ky Fan Inequality that allows us611
to claim existence of maxinf-points in situations when the domain of definition of the612
Ky Fan function is not necessarily compact613
Definition 7 A bivariate function F C times D rarr R with Cand D convex sets in614
f v-biv(Rn+m) is called a Ky Fan function if615
(a) forall y isin D x rarr F(x y) is usc on C 616
(b) forall x isin C y rarr F(x y) is convex on D617
Note that the sets C or D are not required to be compact618
Theorem 8 (lop-limits of Ky Fan functions) The lopsided limit F C times D rarr R of a619
sequence
Fν Cν times Dν rarr RνisinIN of Ky Fan functions in f v-biv(Rn+m) is also a620
Ky Fan function621
Proof For the convexity of y rarr F(x y) let xν isin Cν rarr x isin C be the sequence set622
forth by 3(b) and y0 yλ y1 isin D with yλ = (1 minus λ)y0 + λy1 for λ isin [0 1] In view623
of 3(a) we can choose sequences
y0ν isin Dν rarr y0
y1ν isin Dν rarr y1
such that624
Fν(xν y0ν) rarr F(x y0) and Fν(xν y1ν) rarr F(x y1) Let yλν = (1 minus λ)y0ν +625
λy1ν yλν isin Dν since the functions Fν(x middot) are convex and the sequence yλννisinIN626
certainly converges to yλ For all ν one has627
Fν(xν yλν) le (1 minus λ)Fν(xν y0ν) + λFν(xν y1ν)628
Taking lininf on both sides yields629
F(x yλ) le liminfν Fν(xν yλν) le (1 minus λ)F(x y0) + λF(x y1)630
that establishes the convexity of F(x middot)631
To prove the upper semicontinuity of F with respect to x-variable we show that632
for y isin D633
hypo F(middot y) is the inner set-limit of the hypo Fν(middot yν)634
where the limit is with respect to all sequences yν isin DννisinIN converging to y and635
ν rarr infin This yields the upper semicontinuity since the inner set-limit is always636
closed and a function is usc if and only if its hypograph is closed We have to show637
that if (x α) isin hypo F(middot y) then whenever yν isin Dν rarr y one can find (xν αν) isin638
hypo Fν(middot yν) such that (xν αν) rarr (x α) But that follows immediately from 3(a)639
since we can adjust the αν le Fν(xν yν) so that they converge to α le F(x y) 640
Given a Ky Fan function with compact domain and non-negative on the diagonal641
we have the following important existence result642
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
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ted
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A Jofreacute R J -B Wets
Lemma 2 (Ky Fanrsquos Inequality [7] [4 Theorem 635]) Suppose F C times C rarr R is643
a Ky Fan function with C compact Then the set of maxinf-points of F is a nonempty644
subset of C Moreover if F(x x) ge 0 (on C times C) then for every maxinf-point x of645
C F(x middot) ge 0 on C646
One of the consequences of the lopsided convergence is an extension of the Ky Fanrsquos647
Inequality to the case when it is not possible to apply it directly because one of the648
conditions is not satisfied for example the compactness of the domain However we are649
able to approach the bivariate function F by a sequence FννisinIN defined on compact650
sets Cν This procedure could be useful in many situation where the original maxinf-651
problem is unbounded and then the problem is approached by a family of truncated652
maxinf-problems Such is the case for example when we consider as variables in653
the original problem the multipliers associated to inequality constraints or when the654
original problem is a Walras equilibrium with a positive orthant as consumption set655
in [8] one is precisely confronted with such situations Another simple illustrative656
example follows the statement of the theorem657
Theorem 9 (Extension of Ky Fanrsquos Inequality) Let empty = C sub Rn and F a finite-valued658
bivariate function defined on CtimesC Suppose one can find sequences of compact convex659
setsCν sub R
n
and (finite-valued) Ky Fan functions Fν Cν times Cν rarr RνisinIN lop-660
converging y-tightly to F then every cluster point x of any sequence xν ν isin IN of661
maxinf-points of the Fν is a maxinf-point of F662
Proof Ky Fanrsquos Inequality 2 implies that for all ν the set of maxinf-points of Fν is663
non-empty On the other hand in view of Theorems 8 and 4 any cluster point of such664
maxinf-points will be a maxinf-point of F 665
Example 2 (Extended Ky Fanrsquos Inequality applied) We consider a Ky Fan function666
F(x y) = sin x + (y + 1)minus1 defined on the set [0infin)2 Although667
inf yisin[0infin) F(x y) = sin x668
and the set maxinf-points is not empty we cannot apply Ky Fan Inequality because669
the domain of F is not compact the function F(middot y) is not even sup-compact670
Detail If we consider the functions Fν(x y) = sin x + (y + 1)minus1 on the compact671
domains [0 ν]2 one can apply Ky Fanrsquos Inequality Indeed in this case we have672
inf yisin[0ν) F(x y) = sin x + (ν + 1)minus1673
that converges pointwise and hypo- to sin x and674
argmaxyisin[0ν] inf yisin[0ν] F(x y) = π2 + 2kπ
∣∣ k isin IN
675
Thus xν = π2 and xν = π2 + 2νπ are maxinf-points of the Fν The sequence676
xννisinIN converges to a maxinf-point of F the second sequence xννisinIN does not677
678
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
rrec
ted
proo
f
Variational convergence of bivariate functions
References679
1 Attouch H Variational convergence for functions and operators Applicable Mathematics Series680
Pitman London (1984)681
2 Attouch H Wets RB Convergence des points minsup et de points fixes C R Acad Sci682
Paris 296 657ndash660 (1983)683
3 Attouch H Wets RB A convergence theory for saddle functions Trans Am Math Soc 280684
1ndash41 (1983)685
4 Aubin JP Ekeland I Applied nonlinear analysis Wiley London (1984)686
5 Aubin JP Frankowska H Set-valued analysis Birkhaumluser (1990)687
6 Bagh A Approximation for optimal control problems Lecture at Universidad de Chile Santiago688
(1999)689
7 Fan K A minimax inequality and applications In Shisha O (ed) InequalitiesmdashIII pp 103ndash113690
Academic Dublin (1972)691
8 Jofreacute A Rockafellar R Wets RB A variational inequality scheme for determining an economic692
equilibrium of classical or extended type In Giannessi F Maugeri A (eds) Variational analysis and693
applications pp 553ndash578 Springer New York (2005)694
9 Jofreacute A Wets RB Continuity properties of Walras equilibrium points Ann Oper Res 114 229ndash695
243 (2002)696
10 Jofreacute A Wets RB Variational convergence of bivariate functions Motivating applications II697
(Manuscript) (2006)698
11 Jofreacute A Wets RB Variational convergence of bivariate functions motivating applications I699
(Manuscript) (2004)700
12 Lignola M Morgan J Convergence of marginal functions with dependent con-701
straints Optimization 23 189ndash213 (1992)702
13 Rockafellar R Convex analysis Princeton University Press Princeton (1970)703
14 Rockafellar R Wets RB Variational analysis 2nd edn Springer New York (2004)704
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
rrec
ted
proo
f
Variational convergence of bivariate functionsB
ampW
prin
t
οοο ο ο ο
ο ο
ο ο
ο ο ο ο
οοminus
C
D F(xy)minus
Fig 2 Partition of the domain of a proper bivariate function maxinf framework
lsquolopsidedrsquo-situation and will deal with the lsquosaddle-pointrsquo-situation in the last section344
of the article345
Definitions and results can be stated either in terms of the convergence of maxinf-346
points or minsup-points with some obvious adjustments for signs and terminology347
However its important to know if we are working in a lsquomaxinfrsquo or a lsquominsuprsquo frame-348
work and this is in keeping with the (plain) univariate case where one has to focus349
on either minimization or maximization Because most of the applications we are350
interested in are more naturally formulated in terms of maxinf-problems thats the351
version that will be dealt with in this section We provide at the end of the section352
the necessary translations required to deal with minsup-problems353
Here the term bivariate function always refers to functions defined on the product354
of two non-empty subsets of Rn and R
m respectively6 We write355
biv(Rn+m) = F R
n times Rm rarr R
356
for the class of bivariate functions that are extended real-valued and defined on all of357
Rn times R
m and358
f v-biv(Rn+m) = F C times D rarr R
∣∣ empty = C sub R
n empty = D sub Rm
359
for the class of bivariate functions that are real-valued and defined on the product360
C times D of non-empty subsets of Rn and R
m respectively here its understood that361
Rn+m does not refer to the domain of definition but to the (operational) product space362
that includes C times D363
6 In a follow-up paper we deal with bivariate functions defined on the product of non-empty subsets oftwo topological spaces potentially equipped with different topologies
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
rrec
ted
proo
f
A Jofreacute R J -B Wets
For a bivariate function in biv(Rn+m) or f v-biv(Rn+m) one refers to x as a maxinf-364
point if365
x isin argmaxxisinC
[infyisinD
F(x y)]366
its a minsup-point if367
x isin argminxisinC
[supyisinD
F(x y)]368
C = Rn and D = R
m are not excluded369
Thus for now lets focus on f v-biv(Rn+m) keeping in mind that we are dealing370
with the maxinf case371
Definition 3 (lop-convergence f v-biv) A sequence in f v-biv(Rn+m)
Fν Cν times372
Dν rarr RνisinIN lop-converges or converges lopsided to a function F C times D rarr R373
also in f v-biv(Rn+m) if374
(a) For all xν rarr x with xν isin Cν x isin C and for all y isin D there exists yν rarr y375
with yν isin Dν such that limsupν Fν(xν yν) le F(x y)376
(ainfin) For all xν rarr x with xν isin Cν and x isin C and for all y isin D there exists377
yν rarr y with yν isin Dν such that Fν(xν yν) rarr minusinfin378
(b) For all x isin C there exists xν rarr x with xν isin Cν such that for any sequence379
yν isin DννisinIN liminfν Fν(xν yν) ge F(x y) when the sequence converges to a380
point y isin D and Fν(xν yν) rarr infin when the sequence converges to a point y isin D381
Although a number of properties can be immediately derived from this convergence382
notion cf Theorem 8 for example to obtain the convergence of the maxinf-points383
however we need to require (partial) lsquoy-tightnessrsquo cf Theorem 3 condition (bndasht)384
This (partial) y-tightness condition is new it was inspired by the work of Bagh385
[6] on approximation for optimal control problems A more conventional condition386
that implies y-tightness would be the following (b) holds and there is a compact set387
B sub Rm such that388
forallx isin Rn B sup
y∣∣ Fν(x y) lt infin
389
This last condition suggested in [2] is too restrictive in many applications Moreover390
the use of y-tightness allows for a generalization of Ky Fanrsquos inequality see the next391
section that can be exploited in situations when the domain of definition of the bivariate392
function is not compact393
Now lets turn to the convergence of the marginal functions394
gν = inf yisinDν Fν(middot y) to g = inf yisinD F(middot y)395
in the extended real-valued framework one can find a number of related results in the396
literature see in particular [12]397
123
Journal 10107 MS 0122 CMS 10107_2007_122_Article TYPESET DISK LE CP Disp2007411 Pages 21
unco
rrec
ted
proo
f
Variational convergence of bivariate functions
Theorem 3 (hypo-convergence of the inf-projections) Suppose the sequence398 Fν
νisinIN sub f v-biv(Rn+m) lop-converges to F with condition 3(b) strengthened399
as follows400
(bndasht) not only for all x isin C exist xν isin Cν rarr x such that forall yν isin Dν rarr y401
liminfν Fν(xν yν) ge F(x y) or F(xν yν) rarr infin depending on y belonging or not402
to D but also for any ε gt 0 one can find a compact set Bε possibly depending on403
the sequence xν rarr x such that for all ν larger than some νε404