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Research ArticleLevenberg-Marquardt Method for the EigenvalueComplementarity Problem
Yuan-yuan Chen12 and Yan Gao1
1 School of Management University of Shanghai for Science and Technology Shanghai 200093 China2 College of Mathematics Qingdao University Qingdao 266071 China
Correspondence should be addressed to Yuan-yuan Chen usstchenyuanyuan163com
Received 22 June 2014 Revised 28 August 2014 Accepted 29 August 2014 Published 30 October 2014
Academic Editor Pu-yan Nie
Copyright copy 2014 Y-y Chen and Y Gao This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The eigenvalue complementarity problem (EiCP) is a kind of very useful model which is widely used in the study ofmany problemsin mechanics engineering and economics The EiCP was shown to be equivalent to a special nonlinear complementarity problemor a mathematical programming problem with complementarity constraints The existing methods for solving the EiCP are allnonsmooth methods including nonsmooth or semismooth Newton type methods In this paper we reformulate the EiCP as asystem of continuously differentiable equations and give the Levenberg-Marquardtmethod to solve themUndermild assumptionsthe method is proved globally convergent Finally some numerical results and the extensions of the method are also given Thenumerical experiments highlight the efficiency of the method
1 Introduction
Eigenvalue complementarity problem (EiCP) is proposedin the study of the problems in mechanics engineeringand economics The EiCP is also called cone-constrainedeigenvalue problem in [1ndash4] The EiCP is to find a solutionincluding a scalar and a nonzero vector satisfying a com-plementarity constraint on a closed convex cone The EiCPcan be reformulated to be a special complementarity problemor a mathematical programming optimization problem withcomplementarity constraints and can use nonsmooth orsemismooth Newton type method to solve it such as [5ndash7]The Levenberg-Marquardt method is one of the widely usedmethods in solving optimization problems (see for instance[8ndash15]) Use a trust region strategy to replace the line searchthe Levenberg-Marquardt method is widely considered to bethe progenitor of the trust region method approach for gen-eral unconstrained or constrained optimization problemsThe use of a trust region avoids the weaknesses of GaussNewton method that is its behavior when the Jacobianis rank deficient or nearly so rank deficient On the otherhand we reformulate the EiCP as a system of continuouslydifferentiable equations that is one of the most interesting
themes The advantage of the reformulation is that we solvethe equations with continuously differentiable functions forwhich there are rich powerful solution methods and the-ory analysis including the powerful Levenberg-Marquardtmethod So in this paper we give the Levenberg-Marquardtmethod to solve the EiCP The EiCP which we will consideris the following problem Given the matrix 119860 isin 119877
119899times119899 and thematrix 119861 isin 119877
119899times119899 which are positive definite matrix then weconsider to find a scalar 120582 isin 119877 and a vector 119909 isin 119877
119899
0such that
(119860 minus 120582119861) 119909 ge 0
119909 ge 0
119909119879
(119860 minus 120582119861) 119909 = 0
(1)
This paper is organized as follows In Section 2 we givesome background definitions and known properties Andwe also give the Levenberg-Marquardt method for the EiCPThe global convergence analysis and some discussions of theLevenberg-Marquardt method is also given In Sections 3and 4 we give some numerical results and some extensionsof the method
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 307823 6 pageshttpdxdoiorg1011552014307823
2 The Scientific World Journal
Throughout the paper 119872119899(119877) denotes a real matrix of
order 119899 and 119872119899119898(119877) denotes a real matrix of order 119899 times 119898
0119899= (0 0) and 119890
119899= (1 1)
2 Preliminaries
In this section firstly we reformulate (1) as a system ofcontinuously differentiable equations and give some prelimi-naries used in the followingThenwe propose the Levenberg-Marquardt method for the EiCP
We know that the favorable property of Ψ is that Ψ isa continuously differentiable function on the whole spacealthough 119867 is not a continuously differentiable function ingeneral (see for example [16]) Thus we give the system ofcontinuously differentiable equations for (1) as the followingequations
119865 (120596) = (Ψ (120596)
(119890119899 0) 120596 minus 1
) = 0 (7)
where 119865 119877119899+1 rarr 1198772 which is a continuously differentiable
function In what follows we will give the Levenberg-Marquardt method Denote
Φ (120596) =1
2119865 (120596)
2
(8)
The least-squares formulation of (7) is the following uncon-strained optimization problem
min120596isin119877119899+1
Φ (120596) =1
2119865(120596)
2
(9)
Now we give the Levenberg-Marquardt method forsolving (1) The global convergence result of the method isalso given
Method 1 (the Levenberg-Marquardt method for the EiCP)Given 0 lt 120572 lt 1 0 lt 120573 lt 1 119901 gt 2 120588 gt 0 0 lt 120575 le 2 119875 gt 0120598 gt 0 120596
0isin 119877119899+1 1205830= 119865(120596
0)120575 and 119896 = 0
Step 1 If nablaΦ(120596) le 120598 then stop Otherwise compute 119889119896by
which contradicts with (12) Thus (14) holds So accordingto the definition given in [17] the sequence 119889
119896 is uniformly
gradient related to 120596119896 We complete the proof
Remark 2 In Method 1 we can also use some other linesearch such as the nonmonotone line search The line searchis to find the smallest nonnegative integer119898 such that
Wegive somenumerical experiments for themethod Andwecompare Method 1 with the scaling and projection algorithm(denoted by SPA in [18]) The numerical results indicatethat Method 1 works quite well in practice We considerthe eigenvalue complementarity problems which are alltaken form [3 18] All codes for the method are finished inMATLAB The parameters used in the method are chosen as120588 = 10 119875 = 3 120572 = 01 and 120598 = 10minus4
Example 4 We consider
(119860 minus 120582119861) 119909 ge 0
119909 ge 0
119909119879
(119860 minus 120582119861) 119909 = 0
(24)
where
119861 = (1 0
0 1)
119860 = (8 minus1
3 4)
(25)
By [3] we know that Example 4 has three eigenval-ues Now we consider random initial points to computeExample 4 by Method 1 Numerical results for Example 4 byMethod 1 and SPA method are presented in Table 1
Table 1 shows that Method 1 are able to detect all thesolutions for the small size matrix But the SPA method canonly detect 2 solutions from [3]
Example 5 We consider
(119860 minus 120582119861) 119909 ge 0
119909 ge 0
119909119879
(119860 minus 120582119861) 119909 = 0
(26)
where
119861 = (
1 0 0
0 1 0
0 0 1
)
119860 = (
8 minus1 4
3 4 05
2 minus05 6
)
(27)
From [3] we also know that Example 5 have 9 eigen-values Now we consider random initial points to compute
4 The Scientific World Journal
Table 2
Method 1 120582 119909 SPA 120582 119909
460705 (055617 044898 013390)119879
41340 (0 1 02679)119879
418288 (008023 051776 038374)119879
5 (03333 1 0)119879
503988 (014265 042924 042811)119879
6 (0 0 1)119879
587521 (030666 027220 050890)119879
8 (1 0 0)119879
600946 (012341 036685 050975)119879
sdot sdot sdot sdot sdot sdot
701141 (035296 030395 038277)119879
sdot sdot sdot sdot sdot sdot
800737 (040141 035921 023934)119879
sdot sdot sdot sdot sdot sdot
936479 (047314 0290185 023667)119879
sdot sdot sdot sdot sdot sdot
999838 (057695 013243 029060)119879
sdot sdot sdot sdot sdot sdot
the above Example 5 by Method 1 Numerical results forExample 5 by Method 1 and the SPA method are presentedin Table 2
Table 2 shows that Method 1 is able to detect all thesolutions But the SPA method can only detect 4 Paretoeigenvalues from the analysis of [3]
Example 6 We consider
(119860 minus 120582119861) 119909 ge 0
119909 ge 0
119909119879
(119860 minus 120582119861) 119909 = 0
(28)
where
119861 = (
1 0 0
0 1 0
0 0 1
)
119860 = (
100 106 minus18 minus81
92 158 minus24 minus101
2 44 37 minus7
21 38 0 2
)
(29)
From [3] we also know that Example 6 have 23 eigen-values Now we consider random initial points to computeExample 6 by Method 1 Numerical results for Example 6 byMethod 1 and the SPA method are given in Table 3
Table 3 shows that Method 1 is able to detect all 23solutions But the SPA method can only detect 2 Paretoeigenvalues from [3] The numerical results indicate thatMethod 1 works quite well for the big size EiCP in practice
Discussion In this section we study the numerical behaviorsof Method 1 for solving the Pareto eigenvalue problem TheEiCP problem is very useful in studying the optimizationproblems arising in many areas of the applied mathematicsand mechanics By using the F-B function we reformulatethe EiCP as a system of continuously differentiable equationsThen we use the Levenberg-Marquardt method to solve itFrom the above numerical results we know that Method 1 isvery effective for small size and big size EiCP problems
4 Extensions
41 Bieigenvalue Complementarity Problems (BECP) Wealsocan use Method 1 to solve the bieigenvalue complementarityproblems (denoted by BECP) The BECP is to find (120582 120583) isin119877 times 119877 and (119909 119910) isin 119877119899 0 times 119877119898 0 such that
119909 ge 0 120582119909 minus 119860119909 minus 119861119910 ge 0 119909119879
(120582119909 minus 119860119909 minus 119861119910) = 0
119910 ge 0 120583119910 minus 119862119909 minus 119863119910 ge 0 119910119879
(120583119910 minus 119862119909 minus 119863119910) = 0
(30)
where 119860 isin 119872119899(119877) 119861 isin 119872
and119876(120596) = 120583119910minus119862119909minus119863119910We canwrite the above bieigenvaluecomplementarity problems as 119891(120596) ge 0 119892(120596) ge 0 119875(120596) ge 0119876(120596) ge 0 119891119879(120596)119875(120596) = 0 119892119879(120596)119876(120596) = 0 (119890
119899 0119899 0 0)120596 = 1
and (0119899 119890119899 0 0)120596 = 1 By using F-B function similar to
rewriting the EiCP we can rewrite the above bieigenvaluecomplementarity problems as the following equations
Then we can give the system of continuously differentiableequations for the bieigenvalue complementarity problems asthe following continuously differentiable equations
(
Φ1(120596)
Φ2(120596)
(119890119899 0119899 0 0) 120596 minus 1
(0119899 119890119899 0 0) 120596 minus 1
) = 0 (33)
In the following numerical test the parameters used in themethod are also chosen as 120588 = 10 119875 = 3 120572 = 01 and 120598 =10minus4
Example 7 We consider the BECP where 119860 = 0119863 = 0
119861 = (1 0
0 1)
119862 = (1 0
0 1)
(34)
We use Method 1 to solve the Example 7 Results forExample 7 with random initial point are given in Table 4
42 The Paretian Version We consider using Method 1 tocompute the following Paretian version
119909 ge 0 119872 (120582) 119909 ge 0 119909119879
119872(120582) 119909 = 0 (35)
Table 4
120582 120583 119909 119910
095803 096145 (069993 030006)119879
(069782 030217)119879
096205 097990 (069259 030740)119879
(068925 031074)119879
095574 094739 (048255 051744)119879
(047451 052548)119879
096069 093502 (037373 062626)119879
(037670 062329)119879
094582 104493 (069415 030584)119879
(069598 030401)119879
096200 103264 (046627 053372)119879
(047934 052065)119879
095517 106445 (035869 064130)119879
(037196 062803)119879
095744 105645 (039169 060830)119879
(039707 060292)119879
where 119872 standing for the pencil associated to a finitecollection 119860
0 1198601 119860
119903 of real matrices of order 119899 and
119872(120582) =
119903
sum
119896=0
120582119896
119860119896 (36)
Example 8 We consider the quadratic pencil model where
119872(120582) = 1205822
(
2 0 0
0 6 0
0 0 10
) + 120582(
7 0 0
0 30 0
0 0 20
) + (
minus2 6 0
2 16 3
0 5 0
)
(37)
We use Method 1 to compute Example 8 We present theresults of Example 8 with random initial point in Table 5
6 The Scientific World Journal
Table 5
120582 119909
025315 (097503 001611 000884)119879
minus438990 (000000 100000 000000)119879
minus374482 (095986 002679 001334)119879
minus084644 (039691 042913 017392)119879
minus5000000 (000000 100000 000000)119879
minus060280 (040400 037328 022271)119879
minus366880 (083349 010559 006090)119879
minus187686 (019023 025846 055130)119879
minus198258 (004642 006140 089217)119879
minus001062 (010208 003658 086133)119879
5 Conclusion
In this paper we reformulate the EiCP as a system ofcontinuously differentiable equations and use the Levenberg-Marquardt method to solve them The numerical experi-ments show that our method is a promising method for solv-ing the EiCP The numerical experiments of the extensionsconfirm the efficiency of our method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Science Foundation ofChina (11171221 11101231) Shanghai Leading Academic Dis-cipline (XTKX 2012) and Innovation Program of ShanghaiMunicipal Education Commission (14YZ094)
References
[1] J J Judice M Raydan S S Rosa and S A Santos ldquoOn thesolution of the symmetric eigenvalue complementarity problemby the spectral projected gradient algorithmrdquo Numerical Algo-rithms vol 47 no 4 pp 391ndash407 2008
[2] J J Judice H D Sherali and I M Ribeiro ldquoThe eigenvaluecomplementarity problemrdquo Computational Optimization andApplications vol 37 no 2 pp 139ndash156 2007
[3] C F Ma ldquoThe semismooth and smoothing Newton methodsfor solving Pareto eigenvalue problemrdquo Applied MathematicalModelling vol 36 no 1 pp 279ndash287 2012
[4] S Adly and A Seeger ldquoA nonsmooth algorithm for cone-constrained eigenvalue problemsrdquoComputational Optimizationand Applications vol 49 no 2 pp 299ndash318 2011
[5] S Adly and H Rammal ldquoA new method for solving Paretoeigenvalue complementarity problemsrdquo Computational Opti-mization and Applications vol 55 no 3 pp 703ndash731 2013
[6] J J Judice H D Sherali I M Ribeiro and S S RosaldquoOn the asymmetric eigenvalue complementarity problemrdquoOptimization Methods and Software vol 24 no 4-5 pp 549ndash568 2009
[7] A Pinto da Costa and A Seeger ldquoCone-constrained eigenvalueproblems theory and algorithmsrdquoComputational Optimizationand Applications vol 45 no 1 pp 25ndash57 2010
[8] C Wang Q Liu and C Ma ldquoSmoothing SQP algorithm forsemismooth equations with box constraintsrdquo ComputationalOptimization and Applications vol 55 no 2 pp 399ndash425 2013
[9] F Facchinei and C Kanzow ldquoA nonsmooth inexact Newtonmethod for the solution of large-scale nonlinear complemen-tarity problemsrdquoMathematical Programming vol 76 no 3 pp493ndash512 1997
[10] J Y Fan and Y Y Yuan ldquoOn the quadratic convergenceof the Levenberg-Marquardt method without nonsingularityassumptionrdquo Computing Archives for Scientific Computing vol74 no 1 pp 23ndash39 2005
[11] S-Q Du and Y Gao ldquoThe Levenberg-Marquardt-typemethodsfor a kind of vertical complementarity problemrdquo Journal ofApplied Mathematics vol 2011 Article ID 161853 12 pages 2011
[12] J Y Fan ldquoA Shamanskii-like Levenberg-Marquardt method fornonlinear equationsrdquoComputational Optimization andApplica-tions vol 56 no 1 pp 63ndash80 2013
[13] R Behling and A Fischer ldquoA unified local convergence analysisof inexact constrained Levenberg-Marquardt methodsrdquo Opti-mization Letters vol 6 no 5 pp 927ndash940 2012
[14] K Ueda and N Yamashita ldquoGlobal complexity bound analysisof the Levenberg-Marquardt method for nonsmooth equationsand its application to the nonlinear complementarity problemrdquoJournal of Optimization Theory and Applications vol 152 no 2pp 450ndash467 2012
[15] F Facchinei A Fischer and M Herrich ldquoA family of Newtonmethods for nonsmooth constrained systems with nonisolatedsolutionsrdquo Mathematical Methods of Operations Research vol77 no 3 pp 433ndash443 2013
[16] H Jiang M Fukushima L Qi and D Sun ldquoA trust regionmethod for solving generalized complementarity problemsrdquoSIAM Journal on Optimization vol 8 no 1 pp 140ndash157 1998
[17] D P Bertsekas Constrained Optimization and Lagrange Multi-plier Methods Academic Press New York NY USA 1982
[18] A P da Costa and A Seeger ldquoNumerical resolution of cone-constrained eigenvalue problemsrdquo Computational amp AppliedMathematics vol 28 no 1 pp 37ndash61 2009
Throughout the paper 119872119899(119877) denotes a real matrix of
order 119899 and 119872119899119898(119877) denotes a real matrix of order 119899 times 119898
0119899= (0 0) and 119890
119899= (1 1)
2 Preliminaries
In this section firstly we reformulate (1) as a system ofcontinuously differentiable equations and give some prelimi-naries used in the followingThenwe propose the Levenberg-Marquardt method for the EiCP
We know that the favorable property of Ψ is that Ψ isa continuously differentiable function on the whole spacealthough 119867 is not a continuously differentiable function ingeneral (see for example [16]) Thus we give the system ofcontinuously differentiable equations for (1) as the followingequations
119865 (120596) = (Ψ (120596)
(119890119899 0) 120596 minus 1
) = 0 (7)
where 119865 119877119899+1 rarr 1198772 which is a continuously differentiable
function In what follows we will give the Levenberg-Marquardt method Denote
Φ (120596) =1
2119865 (120596)
2
(8)
The least-squares formulation of (7) is the following uncon-strained optimization problem
min120596isin119877119899+1
Φ (120596) =1
2119865(120596)
2
(9)
Now we give the Levenberg-Marquardt method forsolving (1) The global convergence result of the method isalso given
Method 1 (the Levenberg-Marquardt method for the EiCP)Given 0 lt 120572 lt 1 0 lt 120573 lt 1 119901 gt 2 120588 gt 0 0 lt 120575 le 2 119875 gt 0120598 gt 0 120596
0isin 119877119899+1 1205830= 119865(120596
0)120575 and 119896 = 0
Step 1 If nablaΦ(120596) le 120598 then stop Otherwise compute 119889119896by
which contradicts with (12) Thus (14) holds So accordingto the definition given in [17] the sequence 119889
119896 is uniformly
gradient related to 120596119896 We complete the proof
Remark 2 In Method 1 we can also use some other linesearch such as the nonmonotone line search The line searchis to find the smallest nonnegative integer119898 such that
Wegive somenumerical experiments for themethod Andwecompare Method 1 with the scaling and projection algorithm(denoted by SPA in [18]) The numerical results indicatethat Method 1 works quite well in practice We considerthe eigenvalue complementarity problems which are alltaken form [3 18] All codes for the method are finished inMATLAB The parameters used in the method are chosen as120588 = 10 119875 = 3 120572 = 01 and 120598 = 10minus4
Example 4 We consider
(119860 minus 120582119861) 119909 ge 0
119909 ge 0
119909119879
(119860 minus 120582119861) 119909 = 0
(24)
where
119861 = (1 0
0 1)
119860 = (8 minus1
3 4)
(25)
By [3] we know that Example 4 has three eigenval-ues Now we consider random initial points to computeExample 4 by Method 1 Numerical results for Example 4 byMethod 1 and SPA method are presented in Table 1
Table 1 shows that Method 1 are able to detect all thesolutions for the small size matrix But the SPA method canonly detect 2 solutions from [3]
Example 5 We consider
(119860 minus 120582119861) 119909 ge 0
119909 ge 0
119909119879
(119860 minus 120582119861) 119909 = 0
(26)
where
119861 = (
1 0 0
0 1 0
0 0 1
)
119860 = (
8 minus1 4
3 4 05
2 minus05 6
)
(27)
From [3] we also know that Example 5 have 9 eigen-values Now we consider random initial points to compute
4 The Scientific World Journal
Table 2
Method 1 120582 119909 SPA 120582 119909
460705 (055617 044898 013390)119879
41340 (0 1 02679)119879
418288 (008023 051776 038374)119879
5 (03333 1 0)119879
503988 (014265 042924 042811)119879
6 (0 0 1)119879
587521 (030666 027220 050890)119879
8 (1 0 0)119879
600946 (012341 036685 050975)119879
sdot sdot sdot sdot sdot sdot
701141 (035296 030395 038277)119879
sdot sdot sdot sdot sdot sdot
800737 (040141 035921 023934)119879
sdot sdot sdot sdot sdot sdot
936479 (047314 0290185 023667)119879
sdot sdot sdot sdot sdot sdot
999838 (057695 013243 029060)119879
sdot sdot sdot sdot sdot sdot
the above Example 5 by Method 1 Numerical results forExample 5 by Method 1 and the SPA method are presentedin Table 2
Table 2 shows that Method 1 is able to detect all thesolutions But the SPA method can only detect 4 Paretoeigenvalues from the analysis of [3]
Example 6 We consider
(119860 minus 120582119861) 119909 ge 0
119909 ge 0
119909119879
(119860 minus 120582119861) 119909 = 0
(28)
where
119861 = (
1 0 0
0 1 0
0 0 1
)
119860 = (
100 106 minus18 minus81
92 158 minus24 minus101
2 44 37 minus7
21 38 0 2
)
(29)
From [3] we also know that Example 6 have 23 eigen-values Now we consider random initial points to computeExample 6 by Method 1 Numerical results for Example 6 byMethod 1 and the SPA method are given in Table 3
Table 3 shows that Method 1 is able to detect all 23solutions But the SPA method can only detect 2 Paretoeigenvalues from [3] The numerical results indicate thatMethod 1 works quite well for the big size EiCP in practice
Discussion In this section we study the numerical behaviorsof Method 1 for solving the Pareto eigenvalue problem TheEiCP problem is very useful in studying the optimizationproblems arising in many areas of the applied mathematicsand mechanics By using the F-B function we reformulatethe EiCP as a system of continuously differentiable equationsThen we use the Levenberg-Marquardt method to solve itFrom the above numerical results we know that Method 1 isvery effective for small size and big size EiCP problems
4 Extensions
41 Bieigenvalue Complementarity Problems (BECP) Wealsocan use Method 1 to solve the bieigenvalue complementarityproblems (denoted by BECP) The BECP is to find (120582 120583) isin119877 times 119877 and (119909 119910) isin 119877119899 0 times 119877119898 0 such that
119909 ge 0 120582119909 minus 119860119909 minus 119861119910 ge 0 119909119879
(120582119909 minus 119860119909 minus 119861119910) = 0
119910 ge 0 120583119910 minus 119862119909 minus 119863119910 ge 0 119910119879
(120583119910 minus 119862119909 minus 119863119910) = 0
(30)
where 119860 isin 119872119899(119877) 119861 isin 119872
and119876(120596) = 120583119910minus119862119909minus119863119910We canwrite the above bieigenvaluecomplementarity problems as 119891(120596) ge 0 119892(120596) ge 0 119875(120596) ge 0119876(120596) ge 0 119891119879(120596)119875(120596) = 0 119892119879(120596)119876(120596) = 0 (119890
119899 0119899 0 0)120596 = 1
and (0119899 119890119899 0 0)120596 = 1 By using F-B function similar to
rewriting the EiCP we can rewrite the above bieigenvaluecomplementarity problems as the following equations
Then we can give the system of continuously differentiableequations for the bieigenvalue complementarity problems asthe following continuously differentiable equations
(
Φ1(120596)
Φ2(120596)
(119890119899 0119899 0 0) 120596 minus 1
(0119899 119890119899 0 0) 120596 minus 1
) = 0 (33)
In the following numerical test the parameters used in themethod are also chosen as 120588 = 10 119875 = 3 120572 = 01 and 120598 =10minus4
Example 7 We consider the BECP where 119860 = 0119863 = 0
119861 = (1 0
0 1)
119862 = (1 0
0 1)
(34)
We use Method 1 to solve the Example 7 Results forExample 7 with random initial point are given in Table 4
42 The Paretian Version We consider using Method 1 tocompute the following Paretian version
119909 ge 0 119872 (120582) 119909 ge 0 119909119879
119872(120582) 119909 = 0 (35)
Table 4
120582 120583 119909 119910
095803 096145 (069993 030006)119879
(069782 030217)119879
096205 097990 (069259 030740)119879
(068925 031074)119879
095574 094739 (048255 051744)119879
(047451 052548)119879
096069 093502 (037373 062626)119879
(037670 062329)119879
094582 104493 (069415 030584)119879
(069598 030401)119879
096200 103264 (046627 053372)119879
(047934 052065)119879
095517 106445 (035869 064130)119879
(037196 062803)119879
095744 105645 (039169 060830)119879
(039707 060292)119879
where 119872 standing for the pencil associated to a finitecollection 119860
0 1198601 119860
119903 of real matrices of order 119899 and
119872(120582) =
119903
sum
119896=0
120582119896
119860119896 (36)
Example 8 We consider the quadratic pencil model where
119872(120582) = 1205822
(
2 0 0
0 6 0
0 0 10
) + 120582(
7 0 0
0 30 0
0 0 20
) + (
minus2 6 0
2 16 3
0 5 0
)
(37)
We use Method 1 to compute Example 8 We present theresults of Example 8 with random initial point in Table 5
6 The Scientific World Journal
Table 5
120582 119909
025315 (097503 001611 000884)119879
minus438990 (000000 100000 000000)119879
minus374482 (095986 002679 001334)119879
minus084644 (039691 042913 017392)119879
minus5000000 (000000 100000 000000)119879
minus060280 (040400 037328 022271)119879
minus366880 (083349 010559 006090)119879
minus187686 (019023 025846 055130)119879
minus198258 (004642 006140 089217)119879
minus001062 (010208 003658 086133)119879
5 Conclusion
In this paper we reformulate the EiCP as a system ofcontinuously differentiable equations and use the Levenberg-Marquardt method to solve them The numerical experi-ments show that our method is a promising method for solv-ing the EiCP The numerical experiments of the extensionsconfirm the efficiency of our method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Science Foundation ofChina (11171221 11101231) Shanghai Leading Academic Dis-cipline (XTKX 2012) and Innovation Program of ShanghaiMunicipal Education Commission (14YZ094)
References
[1] J J Judice M Raydan S S Rosa and S A Santos ldquoOn thesolution of the symmetric eigenvalue complementarity problemby the spectral projected gradient algorithmrdquo Numerical Algo-rithms vol 47 no 4 pp 391ndash407 2008
[2] J J Judice H D Sherali and I M Ribeiro ldquoThe eigenvaluecomplementarity problemrdquo Computational Optimization andApplications vol 37 no 2 pp 139ndash156 2007
[3] C F Ma ldquoThe semismooth and smoothing Newton methodsfor solving Pareto eigenvalue problemrdquo Applied MathematicalModelling vol 36 no 1 pp 279ndash287 2012
[4] S Adly and A Seeger ldquoA nonsmooth algorithm for cone-constrained eigenvalue problemsrdquoComputational Optimizationand Applications vol 49 no 2 pp 299ndash318 2011
[5] S Adly and H Rammal ldquoA new method for solving Paretoeigenvalue complementarity problemsrdquo Computational Opti-mization and Applications vol 55 no 3 pp 703ndash731 2013
[6] J J Judice H D Sherali I M Ribeiro and S S RosaldquoOn the asymmetric eigenvalue complementarity problemrdquoOptimization Methods and Software vol 24 no 4-5 pp 549ndash568 2009
[7] A Pinto da Costa and A Seeger ldquoCone-constrained eigenvalueproblems theory and algorithmsrdquoComputational Optimizationand Applications vol 45 no 1 pp 25ndash57 2010
[8] C Wang Q Liu and C Ma ldquoSmoothing SQP algorithm forsemismooth equations with box constraintsrdquo ComputationalOptimization and Applications vol 55 no 2 pp 399ndash425 2013
[9] F Facchinei and C Kanzow ldquoA nonsmooth inexact Newtonmethod for the solution of large-scale nonlinear complemen-tarity problemsrdquoMathematical Programming vol 76 no 3 pp493ndash512 1997
[10] J Y Fan and Y Y Yuan ldquoOn the quadratic convergenceof the Levenberg-Marquardt method without nonsingularityassumptionrdquo Computing Archives for Scientific Computing vol74 no 1 pp 23ndash39 2005
[11] S-Q Du and Y Gao ldquoThe Levenberg-Marquardt-typemethodsfor a kind of vertical complementarity problemrdquo Journal ofApplied Mathematics vol 2011 Article ID 161853 12 pages 2011
[12] J Y Fan ldquoA Shamanskii-like Levenberg-Marquardt method fornonlinear equationsrdquoComputational Optimization andApplica-tions vol 56 no 1 pp 63ndash80 2013
[13] R Behling and A Fischer ldquoA unified local convergence analysisof inexact constrained Levenberg-Marquardt methodsrdquo Opti-mization Letters vol 6 no 5 pp 927ndash940 2012
[14] K Ueda and N Yamashita ldquoGlobal complexity bound analysisof the Levenberg-Marquardt method for nonsmooth equationsand its application to the nonlinear complementarity problemrdquoJournal of Optimization Theory and Applications vol 152 no 2pp 450ndash467 2012
[15] F Facchinei A Fischer and M Herrich ldquoA family of Newtonmethods for nonsmooth constrained systems with nonisolatedsolutionsrdquo Mathematical Methods of Operations Research vol77 no 3 pp 433ndash443 2013
[16] H Jiang M Fukushima L Qi and D Sun ldquoA trust regionmethod for solving generalized complementarity problemsrdquoSIAM Journal on Optimization vol 8 no 1 pp 140ndash157 1998
[17] D P Bertsekas Constrained Optimization and Lagrange Multi-plier Methods Academic Press New York NY USA 1982
[18] A P da Costa and A Seeger ldquoNumerical resolution of cone-constrained eigenvalue problemsrdquo Computational amp AppliedMathematics vol 28 no 1 pp 37ndash61 2009
which contradicts with (12) Thus (14) holds So accordingto the definition given in [17] the sequence 119889
119896 is uniformly
gradient related to 120596119896 We complete the proof
Remark 2 In Method 1 we can also use some other linesearch such as the nonmonotone line search The line searchis to find the smallest nonnegative integer119898 such that
Wegive somenumerical experiments for themethod Andwecompare Method 1 with the scaling and projection algorithm(denoted by SPA in [18]) The numerical results indicatethat Method 1 works quite well in practice We considerthe eigenvalue complementarity problems which are alltaken form [3 18] All codes for the method are finished inMATLAB The parameters used in the method are chosen as120588 = 10 119875 = 3 120572 = 01 and 120598 = 10minus4
Example 4 We consider
(119860 minus 120582119861) 119909 ge 0
119909 ge 0
119909119879
(119860 minus 120582119861) 119909 = 0
(24)
where
119861 = (1 0
0 1)
119860 = (8 minus1
3 4)
(25)
By [3] we know that Example 4 has three eigenval-ues Now we consider random initial points to computeExample 4 by Method 1 Numerical results for Example 4 byMethod 1 and SPA method are presented in Table 1
Table 1 shows that Method 1 are able to detect all thesolutions for the small size matrix But the SPA method canonly detect 2 solutions from [3]
Example 5 We consider
(119860 minus 120582119861) 119909 ge 0
119909 ge 0
119909119879
(119860 minus 120582119861) 119909 = 0
(26)
where
119861 = (
1 0 0
0 1 0
0 0 1
)
119860 = (
8 minus1 4
3 4 05
2 minus05 6
)
(27)
From [3] we also know that Example 5 have 9 eigen-values Now we consider random initial points to compute
4 The Scientific World Journal
Table 2
Method 1 120582 119909 SPA 120582 119909
460705 (055617 044898 013390)119879
41340 (0 1 02679)119879
418288 (008023 051776 038374)119879
5 (03333 1 0)119879
503988 (014265 042924 042811)119879
6 (0 0 1)119879
587521 (030666 027220 050890)119879
8 (1 0 0)119879
600946 (012341 036685 050975)119879
sdot sdot sdot sdot sdot sdot
701141 (035296 030395 038277)119879
sdot sdot sdot sdot sdot sdot
800737 (040141 035921 023934)119879
sdot sdot sdot sdot sdot sdot
936479 (047314 0290185 023667)119879
sdot sdot sdot sdot sdot sdot
999838 (057695 013243 029060)119879
sdot sdot sdot sdot sdot sdot
the above Example 5 by Method 1 Numerical results forExample 5 by Method 1 and the SPA method are presentedin Table 2
Table 2 shows that Method 1 is able to detect all thesolutions But the SPA method can only detect 4 Paretoeigenvalues from the analysis of [3]
Example 6 We consider
(119860 minus 120582119861) 119909 ge 0
119909 ge 0
119909119879
(119860 minus 120582119861) 119909 = 0
(28)
where
119861 = (
1 0 0
0 1 0
0 0 1
)
119860 = (
100 106 minus18 minus81
92 158 minus24 minus101
2 44 37 minus7
21 38 0 2
)
(29)
From [3] we also know that Example 6 have 23 eigen-values Now we consider random initial points to computeExample 6 by Method 1 Numerical results for Example 6 byMethod 1 and the SPA method are given in Table 3
Table 3 shows that Method 1 is able to detect all 23solutions But the SPA method can only detect 2 Paretoeigenvalues from [3] The numerical results indicate thatMethod 1 works quite well for the big size EiCP in practice
Discussion In this section we study the numerical behaviorsof Method 1 for solving the Pareto eigenvalue problem TheEiCP problem is very useful in studying the optimizationproblems arising in many areas of the applied mathematicsand mechanics By using the F-B function we reformulatethe EiCP as a system of continuously differentiable equationsThen we use the Levenberg-Marquardt method to solve itFrom the above numerical results we know that Method 1 isvery effective for small size and big size EiCP problems
4 Extensions
41 Bieigenvalue Complementarity Problems (BECP) Wealsocan use Method 1 to solve the bieigenvalue complementarityproblems (denoted by BECP) The BECP is to find (120582 120583) isin119877 times 119877 and (119909 119910) isin 119877119899 0 times 119877119898 0 such that
119909 ge 0 120582119909 minus 119860119909 minus 119861119910 ge 0 119909119879
(120582119909 minus 119860119909 minus 119861119910) = 0
119910 ge 0 120583119910 minus 119862119909 minus 119863119910 ge 0 119910119879
(120583119910 minus 119862119909 minus 119863119910) = 0
(30)
where 119860 isin 119872119899(119877) 119861 isin 119872
and119876(120596) = 120583119910minus119862119909minus119863119910We canwrite the above bieigenvaluecomplementarity problems as 119891(120596) ge 0 119892(120596) ge 0 119875(120596) ge 0119876(120596) ge 0 119891119879(120596)119875(120596) = 0 119892119879(120596)119876(120596) = 0 (119890
119899 0119899 0 0)120596 = 1
and (0119899 119890119899 0 0)120596 = 1 By using F-B function similar to
rewriting the EiCP we can rewrite the above bieigenvaluecomplementarity problems as the following equations
Then we can give the system of continuously differentiableequations for the bieigenvalue complementarity problems asthe following continuously differentiable equations
(
Φ1(120596)
Φ2(120596)
(119890119899 0119899 0 0) 120596 minus 1
(0119899 119890119899 0 0) 120596 minus 1
) = 0 (33)
In the following numerical test the parameters used in themethod are also chosen as 120588 = 10 119875 = 3 120572 = 01 and 120598 =10minus4
Example 7 We consider the BECP where 119860 = 0119863 = 0
119861 = (1 0
0 1)
119862 = (1 0
0 1)
(34)
We use Method 1 to solve the Example 7 Results forExample 7 with random initial point are given in Table 4
42 The Paretian Version We consider using Method 1 tocompute the following Paretian version
119909 ge 0 119872 (120582) 119909 ge 0 119909119879
119872(120582) 119909 = 0 (35)
Table 4
120582 120583 119909 119910
095803 096145 (069993 030006)119879
(069782 030217)119879
096205 097990 (069259 030740)119879
(068925 031074)119879
095574 094739 (048255 051744)119879
(047451 052548)119879
096069 093502 (037373 062626)119879
(037670 062329)119879
094582 104493 (069415 030584)119879
(069598 030401)119879
096200 103264 (046627 053372)119879
(047934 052065)119879
095517 106445 (035869 064130)119879
(037196 062803)119879
095744 105645 (039169 060830)119879
(039707 060292)119879
where 119872 standing for the pencil associated to a finitecollection 119860
0 1198601 119860
119903 of real matrices of order 119899 and
119872(120582) =
119903
sum
119896=0
120582119896
119860119896 (36)
Example 8 We consider the quadratic pencil model where
119872(120582) = 1205822
(
2 0 0
0 6 0
0 0 10
) + 120582(
7 0 0
0 30 0
0 0 20
) + (
minus2 6 0
2 16 3
0 5 0
)
(37)
We use Method 1 to compute Example 8 We present theresults of Example 8 with random initial point in Table 5
6 The Scientific World Journal
Table 5
120582 119909
025315 (097503 001611 000884)119879
minus438990 (000000 100000 000000)119879
minus374482 (095986 002679 001334)119879
minus084644 (039691 042913 017392)119879
minus5000000 (000000 100000 000000)119879
minus060280 (040400 037328 022271)119879
minus366880 (083349 010559 006090)119879
minus187686 (019023 025846 055130)119879
minus198258 (004642 006140 089217)119879
minus001062 (010208 003658 086133)119879
5 Conclusion
In this paper we reformulate the EiCP as a system ofcontinuously differentiable equations and use the Levenberg-Marquardt method to solve them The numerical experi-ments show that our method is a promising method for solv-ing the EiCP The numerical experiments of the extensionsconfirm the efficiency of our method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Science Foundation ofChina (11171221 11101231) Shanghai Leading Academic Dis-cipline (XTKX 2012) and Innovation Program of ShanghaiMunicipal Education Commission (14YZ094)
References
[1] J J Judice M Raydan S S Rosa and S A Santos ldquoOn thesolution of the symmetric eigenvalue complementarity problemby the spectral projected gradient algorithmrdquo Numerical Algo-rithms vol 47 no 4 pp 391ndash407 2008
[2] J J Judice H D Sherali and I M Ribeiro ldquoThe eigenvaluecomplementarity problemrdquo Computational Optimization andApplications vol 37 no 2 pp 139ndash156 2007
[3] C F Ma ldquoThe semismooth and smoothing Newton methodsfor solving Pareto eigenvalue problemrdquo Applied MathematicalModelling vol 36 no 1 pp 279ndash287 2012
[4] S Adly and A Seeger ldquoA nonsmooth algorithm for cone-constrained eigenvalue problemsrdquoComputational Optimizationand Applications vol 49 no 2 pp 299ndash318 2011
[5] S Adly and H Rammal ldquoA new method for solving Paretoeigenvalue complementarity problemsrdquo Computational Opti-mization and Applications vol 55 no 3 pp 703ndash731 2013
[6] J J Judice H D Sherali I M Ribeiro and S S RosaldquoOn the asymmetric eigenvalue complementarity problemrdquoOptimization Methods and Software vol 24 no 4-5 pp 549ndash568 2009
[7] A Pinto da Costa and A Seeger ldquoCone-constrained eigenvalueproblems theory and algorithmsrdquoComputational Optimizationand Applications vol 45 no 1 pp 25ndash57 2010
[8] C Wang Q Liu and C Ma ldquoSmoothing SQP algorithm forsemismooth equations with box constraintsrdquo ComputationalOptimization and Applications vol 55 no 2 pp 399ndash425 2013
[9] F Facchinei and C Kanzow ldquoA nonsmooth inexact Newtonmethod for the solution of large-scale nonlinear complemen-tarity problemsrdquoMathematical Programming vol 76 no 3 pp493ndash512 1997
[10] J Y Fan and Y Y Yuan ldquoOn the quadratic convergenceof the Levenberg-Marquardt method without nonsingularityassumptionrdquo Computing Archives for Scientific Computing vol74 no 1 pp 23ndash39 2005
[11] S-Q Du and Y Gao ldquoThe Levenberg-Marquardt-typemethodsfor a kind of vertical complementarity problemrdquo Journal ofApplied Mathematics vol 2011 Article ID 161853 12 pages 2011
[12] J Y Fan ldquoA Shamanskii-like Levenberg-Marquardt method fornonlinear equationsrdquoComputational Optimization andApplica-tions vol 56 no 1 pp 63ndash80 2013
[13] R Behling and A Fischer ldquoA unified local convergence analysisof inexact constrained Levenberg-Marquardt methodsrdquo Opti-mization Letters vol 6 no 5 pp 927ndash940 2012
[14] K Ueda and N Yamashita ldquoGlobal complexity bound analysisof the Levenberg-Marquardt method for nonsmooth equationsand its application to the nonlinear complementarity problemrdquoJournal of Optimization Theory and Applications vol 152 no 2pp 450ndash467 2012
[15] F Facchinei A Fischer and M Herrich ldquoA family of Newtonmethods for nonsmooth constrained systems with nonisolatedsolutionsrdquo Mathematical Methods of Operations Research vol77 no 3 pp 433ndash443 2013
[16] H Jiang M Fukushima L Qi and D Sun ldquoA trust regionmethod for solving generalized complementarity problemsrdquoSIAM Journal on Optimization vol 8 no 1 pp 140ndash157 1998
[17] D P Bertsekas Constrained Optimization and Lagrange Multi-plier Methods Academic Press New York NY USA 1982
[18] A P da Costa and A Seeger ldquoNumerical resolution of cone-constrained eigenvalue problemsrdquo Computational amp AppliedMathematics vol 28 no 1 pp 37ndash61 2009
the above Example 5 by Method 1 Numerical results forExample 5 by Method 1 and the SPA method are presentedin Table 2
Table 2 shows that Method 1 is able to detect all thesolutions But the SPA method can only detect 4 Paretoeigenvalues from the analysis of [3]
Example 6 We consider
(119860 minus 120582119861) 119909 ge 0
119909 ge 0
119909119879
(119860 minus 120582119861) 119909 = 0
(28)
where
119861 = (
1 0 0
0 1 0
0 0 1
)
119860 = (
100 106 minus18 minus81
92 158 minus24 minus101
2 44 37 minus7
21 38 0 2
)
(29)
From [3] we also know that Example 6 have 23 eigen-values Now we consider random initial points to computeExample 6 by Method 1 Numerical results for Example 6 byMethod 1 and the SPA method are given in Table 3
Table 3 shows that Method 1 is able to detect all 23solutions But the SPA method can only detect 2 Paretoeigenvalues from [3] The numerical results indicate thatMethod 1 works quite well for the big size EiCP in practice
Discussion In this section we study the numerical behaviorsof Method 1 for solving the Pareto eigenvalue problem TheEiCP problem is very useful in studying the optimizationproblems arising in many areas of the applied mathematicsand mechanics By using the F-B function we reformulatethe EiCP as a system of continuously differentiable equationsThen we use the Levenberg-Marquardt method to solve itFrom the above numerical results we know that Method 1 isvery effective for small size and big size EiCP problems
4 Extensions
41 Bieigenvalue Complementarity Problems (BECP) Wealsocan use Method 1 to solve the bieigenvalue complementarityproblems (denoted by BECP) The BECP is to find (120582 120583) isin119877 times 119877 and (119909 119910) isin 119877119899 0 times 119877119898 0 such that
119909 ge 0 120582119909 minus 119860119909 minus 119861119910 ge 0 119909119879
(120582119909 minus 119860119909 minus 119861119910) = 0
119910 ge 0 120583119910 minus 119862119909 minus 119863119910 ge 0 119910119879
(120583119910 minus 119862119909 minus 119863119910) = 0
(30)
where 119860 isin 119872119899(119877) 119861 isin 119872
and119876(120596) = 120583119910minus119862119909minus119863119910We canwrite the above bieigenvaluecomplementarity problems as 119891(120596) ge 0 119892(120596) ge 0 119875(120596) ge 0119876(120596) ge 0 119891119879(120596)119875(120596) = 0 119892119879(120596)119876(120596) = 0 (119890
119899 0119899 0 0)120596 = 1
and (0119899 119890119899 0 0)120596 = 1 By using F-B function similar to
rewriting the EiCP we can rewrite the above bieigenvaluecomplementarity problems as the following equations
Then we can give the system of continuously differentiableequations for the bieigenvalue complementarity problems asthe following continuously differentiable equations
(
Φ1(120596)
Φ2(120596)
(119890119899 0119899 0 0) 120596 minus 1
(0119899 119890119899 0 0) 120596 minus 1
) = 0 (33)
In the following numerical test the parameters used in themethod are also chosen as 120588 = 10 119875 = 3 120572 = 01 and 120598 =10minus4
Example 7 We consider the BECP where 119860 = 0119863 = 0
119861 = (1 0
0 1)
119862 = (1 0
0 1)
(34)
We use Method 1 to solve the Example 7 Results forExample 7 with random initial point are given in Table 4
42 The Paretian Version We consider using Method 1 tocompute the following Paretian version
119909 ge 0 119872 (120582) 119909 ge 0 119909119879
119872(120582) 119909 = 0 (35)
Table 4
120582 120583 119909 119910
095803 096145 (069993 030006)119879
(069782 030217)119879
096205 097990 (069259 030740)119879
(068925 031074)119879
095574 094739 (048255 051744)119879
(047451 052548)119879
096069 093502 (037373 062626)119879
(037670 062329)119879
094582 104493 (069415 030584)119879
(069598 030401)119879
096200 103264 (046627 053372)119879
(047934 052065)119879
095517 106445 (035869 064130)119879
(037196 062803)119879
095744 105645 (039169 060830)119879
(039707 060292)119879
where 119872 standing for the pencil associated to a finitecollection 119860
0 1198601 119860
119903 of real matrices of order 119899 and
119872(120582) =
119903
sum
119896=0
120582119896
119860119896 (36)
Example 8 We consider the quadratic pencil model where
119872(120582) = 1205822
(
2 0 0
0 6 0
0 0 10
) + 120582(
7 0 0
0 30 0
0 0 20
) + (
minus2 6 0
2 16 3
0 5 0
)
(37)
We use Method 1 to compute Example 8 We present theresults of Example 8 with random initial point in Table 5
6 The Scientific World Journal
Table 5
120582 119909
025315 (097503 001611 000884)119879
minus438990 (000000 100000 000000)119879
minus374482 (095986 002679 001334)119879
minus084644 (039691 042913 017392)119879
minus5000000 (000000 100000 000000)119879
minus060280 (040400 037328 022271)119879
minus366880 (083349 010559 006090)119879
minus187686 (019023 025846 055130)119879
minus198258 (004642 006140 089217)119879
minus001062 (010208 003658 086133)119879
5 Conclusion
In this paper we reformulate the EiCP as a system ofcontinuously differentiable equations and use the Levenberg-Marquardt method to solve them The numerical experi-ments show that our method is a promising method for solv-ing the EiCP The numerical experiments of the extensionsconfirm the efficiency of our method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Science Foundation ofChina (11171221 11101231) Shanghai Leading Academic Dis-cipline (XTKX 2012) and Innovation Program of ShanghaiMunicipal Education Commission (14YZ094)
References
[1] J J Judice M Raydan S S Rosa and S A Santos ldquoOn thesolution of the symmetric eigenvalue complementarity problemby the spectral projected gradient algorithmrdquo Numerical Algo-rithms vol 47 no 4 pp 391ndash407 2008
[2] J J Judice H D Sherali and I M Ribeiro ldquoThe eigenvaluecomplementarity problemrdquo Computational Optimization andApplications vol 37 no 2 pp 139ndash156 2007
[3] C F Ma ldquoThe semismooth and smoothing Newton methodsfor solving Pareto eigenvalue problemrdquo Applied MathematicalModelling vol 36 no 1 pp 279ndash287 2012
[4] S Adly and A Seeger ldquoA nonsmooth algorithm for cone-constrained eigenvalue problemsrdquoComputational Optimizationand Applications vol 49 no 2 pp 299ndash318 2011
[5] S Adly and H Rammal ldquoA new method for solving Paretoeigenvalue complementarity problemsrdquo Computational Opti-mization and Applications vol 55 no 3 pp 703ndash731 2013
[6] J J Judice H D Sherali I M Ribeiro and S S RosaldquoOn the asymmetric eigenvalue complementarity problemrdquoOptimization Methods and Software vol 24 no 4-5 pp 549ndash568 2009
[7] A Pinto da Costa and A Seeger ldquoCone-constrained eigenvalueproblems theory and algorithmsrdquoComputational Optimizationand Applications vol 45 no 1 pp 25ndash57 2010
[8] C Wang Q Liu and C Ma ldquoSmoothing SQP algorithm forsemismooth equations with box constraintsrdquo ComputationalOptimization and Applications vol 55 no 2 pp 399ndash425 2013
[9] F Facchinei and C Kanzow ldquoA nonsmooth inexact Newtonmethod for the solution of large-scale nonlinear complemen-tarity problemsrdquoMathematical Programming vol 76 no 3 pp493ndash512 1997
[10] J Y Fan and Y Y Yuan ldquoOn the quadratic convergenceof the Levenberg-Marquardt method without nonsingularityassumptionrdquo Computing Archives for Scientific Computing vol74 no 1 pp 23ndash39 2005
[11] S-Q Du and Y Gao ldquoThe Levenberg-Marquardt-typemethodsfor a kind of vertical complementarity problemrdquo Journal ofApplied Mathematics vol 2011 Article ID 161853 12 pages 2011
[12] J Y Fan ldquoA Shamanskii-like Levenberg-Marquardt method fornonlinear equationsrdquoComputational Optimization andApplica-tions vol 56 no 1 pp 63ndash80 2013
[13] R Behling and A Fischer ldquoA unified local convergence analysisof inexact constrained Levenberg-Marquardt methodsrdquo Opti-mization Letters vol 6 no 5 pp 927ndash940 2012
[14] K Ueda and N Yamashita ldquoGlobal complexity bound analysisof the Levenberg-Marquardt method for nonsmooth equationsand its application to the nonlinear complementarity problemrdquoJournal of Optimization Theory and Applications vol 152 no 2pp 450ndash467 2012
[15] F Facchinei A Fischer and M Herrich ldquoA family of Newtonmethods for nonsmooth constrained systems with nonisolatedsolutionsrdquo Mathematical Methods of Operations Research vol77 no 3 pp 433ndash443 2013
[16] H Jiang M Fukushima L Qi and D Sun ldquoA trust regionmethod for solving generalized complementarity problemsrdquoSIAM Journal on Optimization vol 8 no 1 pp 140ndash157 1998
[17] D P Bertsekas Constrained Optimization and Lagrange Multi-plier Methods Academic Press New York NY USA 1982
[18] A P da Costa and A Seeger ldquoNumerical resolution of cone-constrained eigenvalue problemsrdquo Computational amp AppliedMathematics vol 28 no 1 pp 37ndash61 2009
Then we can give the system of continuously differentiableequations for the bieigenvalue complementarity problems asthe following continuously differentiable equations
(
Φ1(120596)
Φ2(120596)
(119890119899 0119899 0 0) 120596 minus 1
(0119899 119890119899 0 0) 120596 minus 1
) = 0 (33)
In the following numerical test the parameters used in themethod are also chosen as 120588 = 10 119875 = 3 120572 = 01 and 120598 =10minus4
Example 7 We consider the BECP where 119860 = 0119863 = 0
119861 = (1 0
0 1)
119862 = (1 0
0 1)
(34)
We use Method 1 to solve the Example 7 Results forExample 7 with random initial point are given in Table 4
42 The Paretian Version We consider using Method 1 tocompute the following Paretian version
119909 ge 0 119872 (120582) 119909 ge 0 119909119879
119872(120582) 119909 = 0 (35)
Table 4
120582 120583 119909 119910
095803 096145 (069993 030006)119879
(069782 030217)119879
096205 097990 (069259 030740)119879
(068925 031074)119879
095574 094739 (048255 051744)119879
(047451 052548)119879
096069 093502 (037373 062626)119879
(037670 062329)119879
094582 104493 (069415 030584)119879
(069598 030401)119879
096200 103264 (046627 053372)119879
(047934 052065)119879
095517 106445 (035869 064130)119879
(037196 062803)119879
095744 105645 (039169 060830)119879
(039707 060292)119879
where 119872 standing for the pencil associated to a finitecollection 119860
0 1198601 119860
119903 of real matrices of order 119899 and
119872(120582) =
119903
sum
119896=0
120582119896
119860119896 (36)
Example 8 We consider the quadratic pencil model where
119872(120582) = 1205822
(
2 0 0
0 6 0
0 0 10
) + 120582(
7 0 0
0 30 0
0 0 20
) + (
minus2 6 0
2 16 3
0 5 0
)
(37)
We use Method 1 to compute Example 8 We present theresults of Example 8 with random initial point in Table 5
6 The Scientific World Journal
Table 5
120582 119909
025315 (097503 001611 000884)119879
minus438990 (000000 100000 000000)119879
minus374482 (095986 002679 001334)119879
minus084644 (039691 042913 017392)119879
minus5000000 (000000 100000 000000)119879
minus060280 (040400 037328 022271)119879
minus366880 (083349 010559 006090)119879
minus187686 (019023 025846 055130)119879
minus198258 (004642 006140 089217)119879
minus001062 (010208 003658 086133)119879
5 Conclusion
In this paper we reformulate the EiCP as a system ofcontinuously differentiable equations and use the Levenberg-Marquardt method to solve them The numerical experi-ments show that our method is a promising method for solv-ing the EiCP The numerical experiments of the extensionsconfirm the efficiency of our method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Science Foundation ofChina (11171221 11101231) Shanghai Leading Academic Dis-cipline (XTKX 2012) and Innovation Program of ShanghaiMunicipal Education Commission (14YZ094)
References
[1] J J Judice M Raydan S S Rosa and S A Santos ldquoOn thesolution of the symmetric eigenvalue complementarity problemby the spectral projected gradient algorithmrdquo Numerical Algo-rithms vol 47 no 4 pp 391ndash407 2008
[2] J J Judice H D Sherali and I M Ribeiro ldquoThe eigenvaluecomplementarity problemrdquo Computational Optimization andApplications vol 37 no 2 pp 139ndash156 2007
[3] C F Ma ldquoThe semismooth and smoothing Newton methodsfor solving Pareto eigenvalue problemrdquo Applied MathematicalModelling vol 36 no 1 pp 279ndash287 2012
[4] S Adly and A Seeger ldquoA nonsmooth algorithm for cone-constrained eigenvalue problemsrdquoComputational Optimizationand Applications vol 49 no 2 pp 299ndash318 2011
[5] S Adly and H Rammal ldquoA new method for solving Paretoeigenvalue complementarity problemsrdquo Computational Opti-mization and Applications vol 55 no 3 pp 703ndash731 2013
[6] J J Judice H D Sherali I M Ribeiro and S S RosaldquoOn the asymmetric eigenvalue complementarity problemrdquoOptimization Methods and Software vol 24 no 4-5 pp 549ndash568 2009
[7] A Pinto da Costa and A Seeger ldquoCone-constrained eigenvalueproblems theory and algorithmsrdquoComputational Optimizationand Applications vol 45 no 1 pp 25ndash57 2010
[8] C Wang Q Liu and C Ma ldquoSmoothing SQP algorithm forsemismooth equations with box constraintsrdquo ComputationalOptimization and Applications vol 55 no 2 pp 399ndash425 2013
[9] F Facchinei and C Kanzow ldquoA nonsmooth inexact Newtonmethod for the solution of large-scale nonlinear complemen-tarity problemsrdquoMathematical Programming vol 76 no 3 pp493ndash512 1997
[10] J Y Fan and Y Y Yuan ldquoOn the quadratic convergenceof the Levenberg-Marquardt method without nonsingularityassumptionrdquo Computing Archives for Scientific Computing vol74 no 1 pp 23ndash39 2005
[11] S-Q Du and Y Gao ldquoThe Levenberg-Marquardt-typemethodsfor a kind of vertical complementarity problemrdquo Journal ofApplied Mathematics vol 2011 Article ID 161853 12 pages 2011
[12] J Y Fan ldquoA Shamanskii-like Levenberg-Marquardt method fornonlinear equationsrdquoComputational Optimization andApplica-tions vol 56 no 1 pp 63ndash80 2013
[13] R Behling and A Fischer ldquoA unified local convergence analysisof inexact constrained Levenberg-Marquardt methodsrdquo Opti-mization Letters vol 6 no 5 pp 927ndash940 2012
[14] K Ueda and N Yamashita ldquoGlobal complexity bound analysisof the Levenberg-Marquardt method for nonsmooth equationsand its application to the nonlinear complementarity problemrdquoJournal of Optimization Theory and Applications vol 152 no 2pp 450ndash467 2012
[15] F Facchinei A Fischer and M Herrich ldquoA family of Newtonmethods for nonsmooth constrained systems with nonisolatedsolutionsrdquo Mathematical Methods of Operations Research vol77 no 3 pp 433ndash443 2013
[16] H Jiang M Fukushima L Qi and D Sun ldquoA trust regionmethod for solving generalized complementarity problemsrdquoSIAM Journal on Optimization vol 8 no 1 pp 140ndash157 1998
[17] D P Bertsekas Constrained Optimization and Lagrange Multi-plier Methods Academic Press New York NY USA 1982
[18] A P da Costa and A Seeger ldquoNumerical resolution of cone-constrained eigenvalue problemsrdquo Computational amp AppliedMathematics vol 28 no 1 pp 37ndash61 2009
In this paper we reformulate the EiCP as a system ofcontinuously differentiable equations and use the Levenberg-Marquardt method to solve them The numerical experi-ments show that our method is a promising method for solv-ing the EiCP The numerical experiments of the extensionsconfirm the efficiency of our method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work is supported by National Science Foundation ofChina (11171221 11101231) Shanghai Leading Academic Dis-cipline (XTKX 2012) and Innovation Program of ShanghaiMunicipal Education Commission (14YZ094)
References
[1] J J Judice M Raydan S S Rosa and S A Santos ldquoOn thesolution of the symmetric eigenvalue complementarity problemby the spectral projected gradient algorithmrdquo Numerical Algo-rithms vol 47 no 4 pp 391ndash407 2008
[2] J J Judice H D Sherali and I M Ribeiro ldquoThe eigenvaluecomplementarity problemrdquo Computational Optimization andApplications vol 37 no 2 pp 139ndash156 2007
[3] C F Ma ldquoThe semismooth and smoothing Newton methodsfor solving Pareto eigenvalue problemrdquo Applied MathematicalModelling vol 36 no 1 pp 279ndash287 2012
[4] S Adly and A Seeger ldquoA nonsmooth algorithm for cone-constrained eigenvalue problemsrdquoComputational Optimizationand Applications vol 49 no 2 pp 299ndash318 2011
[5] S Adly and H Rammal ldquoA new method for solving Paretoeigenvalue complementarity problemsrdquo Computational Opti-mization and Applications vol 55 no 3 pp 703ndash731 2013
[6] J J Judice H D Sherali I M Ribeiro and S S RosaldquoOn the asymmetric eigenvalue complementarity problemrdquoOptimization Methods and Software vol 24 no 4-5 pp 549ndash568 2009
[7] A Pinto da Costa and A Seeger ldquoCone-constrained eigenvalueproblems theory and algorithmsrdquoComputational Optimizationand Applications vol 45 no 1 pp 25ndash57 2010
[8] C Wang Q Liu and C Ma ldquoSmoothing SQP algorithm forsemismooth equations with box constraintsrdquo ComputationalOptimization and Applications vol 55 no 2 pp 399ndash425 2013
[9] F Facchinei and C Kanzow ldquoA nonsmooth inexact Newtonmethod for the solution of large-scale nonlinear complemen-tarity problemsrdquoMathematical Programming vol 76 no 3 pp493ndash512 1997
[10] J Y Fan and Y Y Yuan ldquoOn the quadratic convergenceof the Levenberg-Marquardt method without nonsingularityassumptionrdquo Computing Archives for Scientific Computing vol74 no 1 pp 23ndash39 2005
[11] S-Q Du and Y Gao ldquoThe Levenberg-Marquardt-typemethodsfor a kind of vertical complementarity problemrdquo Journal ofApplied Mathematics vol 2011 Article ID 161853 12 pages 2011
[12] J Y Fan ldquoA Shamanskii-like Levenberg-Marquardt method fornonlinear equationsrdquoComputational Optimization andApplica-tions vol 56 no 1 pp 63ndash80 2013
[13] R Behling and A Fischer ldquoA unified local convergence analysisof inexact constrained Levenberg-Marquardt methodsrdquo Opti-mization Letters vol 6 no 5 pp 927ndash940 2012
[14] K Ueda and N Yamashita ldquoGlobal complexity bound analysisof the Levenberg-Marquardt method for nonsmooth equationsand its application to the nonlinear complementarity problemrdquoJournal of Optimization Theory and Applications vol 152 no 2pp 450ndash467 2012
[15] F Facchinei A Fischer and M Herrich ldquoA family of Newtonmethods for nonsmooth constrained systems with nonisolatedsolutionsrdquo Mathematical Methods of Operations Research vol77 no 3 pp 433ndash443 2013
[16] H Jiang M Fukushima L Qi and D Sun ldquoA trust regionmethod for solving generalized complementarity problemsrdquoSIAM Journal on Optimization vol 8 no 1 pp 140ndash157 1998
[17] D P Bertsekas Constrained Optimization and Lagrange Multi-plier Methods Academic Press New York NY USA 1982
[18] A P da Costa and A Seeger ldquoNumerical resolution of cone-constrained eigenvalue problemsrdquo Computational amp AppliedMathematics vol 28 no 1 pp 37ndash61 2009