-
Col. Gen. Stabilization using Dual Smoothing:Theory &
Practice
Colgen Workshop, June 2012
J. Han (1), P. Pesneau (2), A. Pessoa (3),R. Sadykov (1), E.
Uchoa (3), F. Vanderbeck (1)
(1) INRIA Bordeaux-Sud-Ouest, team RealOpt
(2) Université de Bordeaux, Institut de Mathématiques
(3) LOGIS , Universidade Federal Fluminense
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Outline
1 Column Generation & Cut Separation in the Dual
2 Stabilization techniques
3 Numerical Analysis
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Column Generation & Cut Separation in the Dual
Problem decomposition
Assume a bounded integer integer problem:
[F]≡ min{cx :x ∈ Y ≡ Ax ≥ ax ∈ Z ≡ Bx ≥ b
x ∈ INn
Let X := Y ∩ Z . Assume that subproblem
[SP]≡ min{cx : x ∈ Z} (1)
is “relatively easy” to solve compared to problem [F]. Then,
Z = {zq}q∈Q= {x ∈ INn : x =
∑
q∈Qzqλq,∑
q∈Qλq = 1; λq ≥ 0 ∀q ∈ Q}
and
conv(Z) = {x ∈ IRn+ :∑
q∈Qzqλq,∑
q∈Qλq = 1, λq ≥ 0 q ∈ Q}
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
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Column Generation & Cut Separation in the Dual
Lagrangian Relaxation & Duality
L(π) := minq∈Q{c zq +π(a− Azq)}
[LD] := maxπ∈IRm+
minq∈Q{c zq +π(a− Azq)}
π
η
π
η
L(π)
π
η
L(π)
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
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Column Generation & Cut Separation in the Dual
Lagrangian Dual as an LP
[LD] ≡ maxπ∈IRm+
minq∈Q{π a + (c−πA)zq};
≡ max{η,η≤ czq +π(a− Azq) q ∈ Q,π ∈ IRm+,η ∈ IR
1};
≡ min{∑
q∈Q(czq)λq,
∑
q∈Q(Azq)λq ≥ a,
∑
q∈Qλq = 1,
λq ≥ 0 q ∈ Q};≡ min{cx : Ax ≥ a, x ∈ conv(Z) }.
π
η
L(π)
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
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Column Generation & Cut Separation in the Dual
Dantzig-Wolfe Reformulation & Restricted Master
min∑
q∈Q(cx)λq
∑
q∈Q(Azq)λq ≥ a
∑
q∈Qλq = 1
λq ∈ {0, 1} ∀q ∈ Q .
[Mt ]≡ min{∑
q∈Qtczqλq :∑
q∈QtAzqλq ≥ a;∑
q∈Qtλq = 1;λq ≥ 0, q ∈ Qt}
[DMt ]≡ max{η : π(Azq − a) +η≤ czq, q ∈ Qt ;π ∈ IRm+;η ∈
IR1}
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
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Column Generation & Cut Separation in the Dual
Restricted Master, Dual Polyhedra, & Pricing Oracle
[Mt ]≡ min{cx : Ax ≥ a, x ∈ conv({zq}q∈Qt )}.
Lt() : π→ Lt(π) =minq∈Qt {πa + (c−πA)zq};
Solving [LSP(πt )] yields:1 most neg. red. cost col. for [Mt ]2
most violated constr. for [DMt ]3 correct value L() at point πt
π
η
L(π)
(πt ,ηt)
π
η
L(π)
(πt ,ηt)
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
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Column Generation & Cut Separation in the Dual
Restricted Master, Dual Polyhedra, & Pricing Oracle
[Mt ]≡ min{cx : Ax ≥ a, x ∈ conv({zq}q∈Qt )}.
Lt() : π→ Lt(π) =minq∈Qt {πa + (c−πA)zq};
Solving [LSP(πt )] yields:1 most neg. red. cost col. for [Mt ]2
most violated constr. for [DMt ]3 correct value L() at point πt
π
η
L(π)
(πt ,ηt)
π
η
L(π)
(πt ,ηt)
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
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Column Generation & Cut Separation in the Dual
Dual Polyhedra: Outer and Inner approximations
π
η
L(π)
(πt ,ηt)(π̂, L̂)
π
η
L(π)
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
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Column Generation & Cut Separation in the Dual
Cut Separation Strategies in the Dual
Column generation for a master program≡ cut generation for the
dual master
⇓
Cutting plane “strategies” translate into in col. gen.
“stabilization”
In-Out separation [BenAmeurNeto07, FischettiSalvagnin10]
Central point cutting strategy [GoffinVial, LeePark11]
Lexicographic Simplex [ZanetteFischettiBalas11]
...
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Column Generation & Cut Separation in the Dual
Cut Separation Strategies in the Dual
Column generation for a master program≡ cut generation for the
dual master
⇓
Cutting plane “strategies” translate into in col. gen.
“stabilization”
In-Out separation [BenAmeurNeto07, FischettiSalvagnin10]
Central point cutting strategy [GoffinVial, LeePark11]
Lexicographic Simplex [ZanetteFischettiBalas11]
...
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Column Generation & Cut Separation in the Dual
Methods to Solve the Lagrangian Dual
A sequence of candidate dual solutions
{πt}t → π∗ ∈ Π∗
A sequence of candidate primal solutions (a by-product to
proveoptimality)
{x t}t → x∗ ∈ X ∗
Oracle: z t ← argminx∈Z {(c−πt−1A)x}.
1 Ascent methods:SubgradientVolumeConjugate Sub-gradient
2 Polyhedral methods:KelleyBundleACCPM
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Column Generation & Cut Separation in the Dual
Methods to Solve the Lagrangian Dual
A sequence of candidate dual solutions
{πt}t → π∗ ∈ Π∗
A sequence of candidate primal solutions (a by-product to
proveoptimality)
{x t}t → x∗ ∈ X ∗
Oracle: z t ← argminx∈Z {(c−πt−1A)x}.
1 Ascent methods:SubgradientVolumeConjugate Sub-gradient
2 Polyhedral methods:KelleyBundleACCPM
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Stabilization techniques
Convergence
844
0 100 200 300 400 500 600 700 800 900 1000
0
84
169
253
338
422
506
591
675
760
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Stabilization techniques
Convergence
{πt}t → π∗ ∈Π∗
Dual oscillations: πt jump erratically (bang-bang), {L(πt)}
nonmonotic (yo-yo) and possibly
||πt −π∗||> ||πt−1−π∗||
Tailing-off effect: towards the end, added inequalities
yieldmarginal improvements / step sizes are very small.
Primal degeneracy / multiple dual optima:fewer non zero λq than
master constraints;dual system with fewer constraints than
variables;cuts co-linear with objective (inherent to cutting plane
procedures).
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Stabilization techniques
Stabilization Techniques
Penalty functions to drive the optimization towards π̂:
πt := argmaxπ∈IRm+{Lt(π)− Ŝ(π)}
Dual price smoothing (In-Out separation)
[Wentges97] π̃t = απ̂+ (1−α)πt
[Neame99] π̃t = απ̃t−1 + (1−α)πt
Dual price centralization (central point separation)Analytic
center (of trust polyhedra)Chebyshev centerOptimal face center
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Stabilization techniques
Stabilization Techniques
Penalty functions to drive the optimization towards π̂:
πt := argmaxπ∈IRm+{Lt(π)− Ŝ(π)}
Dual price smoothing (In-Out separation)
[Wentges97] π̃t = απ̂+ (1−α)πt
[Neame99] π̃t = απ̃t−1 + (1−α)πt
Dual price centralization (central point separation)Analytic
center (of trust polyhedra)Chebyshev centerOptimal face center
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Stabilization techniques
Dual Price Smoothing ≡ In-Out Separation
π̃t = απ̂+ (1−α)πt ↔ (Wentges ’s rule)
(πin,ηin) := (π̂, L̂) (2)
(πout,ηout) := (πt ,ηt) . (3)
(πsep,ηsep) := α (πin,ηin) + (1−α) (πout,ηout) . (4)
OUT
SEP
IN
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Stabilization techniques
Dual Price Smoothing ≡ In-Out Separation
(πin,ηin) :=
¨
(π̂, L̂) under Wentges ’s rule.(π̃t−1, L(π̃t−1)) under Neame’s
rule,
(5)
(πout,ηout) := (πt ,ηt) . (6)
(πsep,ηsep) := α (πin,ηin) + (1−α) (πout,ηout) . (7)
Oracle: z t ← argminx∈Z {(c−πsepA)x}
OUT
SEP
IN
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Stabilization techniques
In-Out Separation
Case A: SEP is cut, so is OUTCase B: SEP is not cut, but
OUT is cutCase C: neither SEP nor OUT
is cut→ “mis-price”
OUT
SEP
IN
OUT
SEP
IN
OUT
SEPIN
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Stabilization techniques
Properties
1 If SEP is cut by z t , OUT is cut.2 Otherwise,
a) SEP defines the next IN point.b) (L(πsep)−ηin) ≥
(1−α) (ηout −ηin) , i.e.,
ηout − L(πsep) ≤ α (ηout −ηin)
c) The OUT point might be cut.d) Otherwise, πt = πt−1 = πout
and
||π̃t −πout||= α ||π̃t−1 −πout||
In case of “mis-price”
πout πsep πin π
η
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Stabilization techniques
Dual Smoothing: Termination of the Algorithm
The number of columns that can be generated is finite.
A column, once added, cannot be regenerated: the
associatedconstraint is satisfied by both IN and OUT points (hence
by SEP).
However, there are iterations where no colunms are generated,and
the OUT point remains unchanged→ mis-priceThus, on needs to prove
convergence of a subsequence ofmis-price iterations, starting with
πin = π̃0. Then,π̃t+1 = αt π̃
t + (1−αt)πout . Hence,
||π̃t+1−πout ||= αt ||π̃t −πout ||= . . . = Πtτ=0ατ ||π̃0−πout
||
if α= 0.8π̃0 π̃1 π̃2 π̃3 πout
.2 .16 .128
At iteration t , ||π̃0−πout || is cut by a factor
(1−Πtτ=0ατ).
BEWARE: convergence is only asymptotic for a constant 0< α
< 1.Colgen Workshop, June 2012 Col. Gen. Stabilization using
Dual Smoothing: Theory & Practice
-
Stabilization techniques
Dual Smoothing: Termination of the Algorithm
The number of columns that can be generated is finite.
A column, once added, cannot be regenerated: the
associatedconstraint is satisfied by both IN and OUT points (hence
by SEP).
However, there are iterations where no colunms are generated,and
the OUT point remains unchanged→ mis-priceThus, on needs to prove
convergence of a subsequence ofmis-price iterations, starting with
πin = π̃0. Then,π̃t+1 = αt π̃
t + (1−αt)πout . Hence,
||π̃t+1−πout ||= αt ||π̃t −πout ||= . . . = Πtτ=0ατ ||π̃0−πout
||
if α= 0.8π̃0 π̃1 π̃2 π̃3 πout
.2 .16 .128
At iteration t , ||π̃0−πout || is cut by a factor
(1−Πtτ=0ατ).
BEWARE: convergence is only asymptotic for a constant 0< α
< 1.Colgen Workshop, June 2012 Col. Gen. Stabilization using
Dual Smoothing: Theory & Practice
-
Stabilization techniques
Dual Smoothing: Termination of the Algorithm
The number of columns that can be generated is finite.
A column, once added, cannot be regenerated: the
associatedconstraint is satisfied by both IN and OUT points (hence
by SEP).
However, there are iterations where no colunms are generated,and
the OUT point remains unchanged→ mis-priceThus, on needs to prove
convergence of a subsequence ofmis-price iterations, starting with
πin = π̃0. Then,π̃t+1 = αt π̃
t + (1−αt)πout . Hence,
||π̃t+1−πout ||= αt ||π̃t −πout ||= . . . = Πtτ=0ατ ||π̃0−πout
||
if α= 0.8π̃0 π̃1 π̃2 π̃3 πout
.2 .16 .128
At iteration t , ||π̃0−πout || is cut by a factor
(1−Πtτ=0ατ).
BEWARE: convergence is only asymptotic for a constant 0< α
< 1.Colgen Workshop, June 2012 Col. Gen. Stabilization using
Dual Smoothing: Theory & Practice
-
Stabilization techniques
Dual Smoothing: Termination of the Algorithm
The number of columns that can be generated is finite.
A column, once added, cannot be regenerated: the
associatedconstraint is satisfied by both IN and OUT points (hence
by SEP).
However, there are iterations where no colunms are generated,and
the OUT point remains unchanged→ mis-priceThus, on needs to prove
convergence of a subsequence ofmis-price iterations, starting with
πin = π̃0. Then,π̃t+1 = αt π̃
t + (1−αt)πout . Hence,
||π̃t+1−πout ||= αt ||π̃t −πout ||= . . . = Πtτ=0ατ ||π̃0−πout
||
if α= 0.8π̃0 π̃1 π̃2 π̃3 πout
.2 .16 .128
At iteration t , ||π̃0−πout || is cut by a factor
(1−Πtτ=0ατ).
BEWARE: convergence is only asymptotic for a constant 0< α
< 1.Colgen Workshop, June 2012 Col. Gen. Stabilization using
Dual Smoothing: Theory & Practice
-
Stabilization techniques
Dual Smoothing: Termination of the Algorithm
The number of columns that can be generated is finite.
A column, once added, cannot be regenerated: the
associatedconstraint is satisfied by both IN and OUT points (hence
by SEP).
However, there are iterations where no colunms are generated,and
the OUT point remains unchanged→ mis-priceThus, on needs to prove
convergence of a subsequence ofmis-price iterations, starting with
πin = π̃0. Then,π̃t+1 = αt π̃
t + (1−αt)πout . Hence,
||π̃t+1−πout ||= αt ||π̃t −πout ||= . . . = Πtτ=0ατ ||π̃0−πout
||
if α= 0.8π̃0 π̃1 π̃2 π̃3 πout
.2 .16 .128
At iteration t , ||π̃0−πout || is cut by a factor
(1−Πtτ=0ατ).
BEWARE: convergence is only asymptotic for a constant 0< α
< 1.Colgen Workshop, June 2012 Col. Gen. Stabilization using
Dual Smoothing: Theory & Practice
-
Stabilization techniques
Dual Smoothing: Termination “in Theory” vs “in Practice”
In case of mis-pricing, one observes a sequence of iterations in
which:||π̃t+1 −πout ||= αt ||π̃t −πout ||||ηt+1 − L(π̃t+1)|| ≤ αt
||ηt − L(π̃t)||
Hence, as (1−Πtτ=0ατ)→ 1,
π̃t → πoutL(π̃t)→ ηout
After a finite number of iterations, ||ηt − L(π̃t)|| ≤ ε and the
master isconsidered as optimized. As there is a finite number of
possiblevalues for L(π̃t) and ηt ,
∃ε : ||ηt − L(π̃t)|| ≤ ε ⇒ ηt = L(π̃t)
which proves finite convergence in theory
[BenAmeurNeto07,Wentges97].
In practice, for finite convergence, one better choose ατ :
(1−Πtτ=0ατ) ≥ 1 for a finite t
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Stabilization techniques
Dual Smoothing: α-schedule for finite convergence in
practice
Algorithm for handling a mis-pricing sequence
Step 0 k ← 1Step 1 β ← [1− k ∗ (1−α)]+
Step 2 πsep = β π0 + (1− β)πout
Step 3 k ← k + 1Step 4 call the oracle on πsep
Step 5 if mis-pricing occurs, goto Step 1else, got Step 0.
if α= 0.8π̃0 π̃1 π̃2 π̃3 πout
.2 .2 .2
I.e; we chose ατ : (1−Πkτ=0ατ) = k ∗ (1−α). Hence,(1−Πkτ=0ατ)≥ 1
after k =
l
1(1−α)
m
iterations, and smoothing stops
with β = 0.
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
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Stabilization techniques
Dual Smoothing: Implementation in Practice
Turn-off smoothing during HEADING-IN Phase.
Trust πout during TAILING-OFF Phase: small α.
Decreasing α as the algorithm converges: f.i., record
decreasedα-value in mis-pricing sequence.
844
0 100 200 300 400 500 600 700 800 900 1000
0
84
169
253
338
422
506
591
675
760
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Numerical Analysis
Sensivity to static α: Machine Scheduling
12 jobs per machines(25 jobs / 2 machines ; 50 jobs / 4
machines)
0 0.2 0.4 0.6 0.8 0.95
200
250
300
350
smoothing parameter (α)
Master iterationsSubproblem calls
0 0.2 0.4 0.6 0.8 0.95
0.4
0.6
0.8
1
smoothing parameter (α)
Time (sec.)
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Numerical Analysis
Sensivity to static α: Machine Scheduling
25 jobs per machines25 jobs / 1 machine ; 50 jobs / 2 machines;
100 jobs / 4 machines
0 0.2 0.4 0.6 0.8 0.95300
400
500
600
700
800
smoothing parameter (α)
Master iterationsSubproblem calls
0 0.2 0.4 0.6 0.8 0.95
4
6
8
10
12
smoothing parameter (α)
Time (sec.)
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Numerical Analysis
Sensivity to static α: Machine Scheduling
50 jobs per machines50 jobs / 1 machine ; 100 jobs / 2 machines;
200 jobs / 4 machines
0 0.2 0.4 0.6 0.8 0.95500
1,000
1,500
2,000
2,500
3,000
smoothing parameter (α)
Master iterationsSubproblem calls
0 0.2 0.4 0.6 0.8 0.950
50
100
150
200
smoothing parameter (α)
Time (sec.)
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Numerical Analysis
Sensivity to static α: Machine Scheduling
100 jobs per machines100 jobs / 1 machine
0 0.2 0.4 0.6 0.8 0.95
103
104
smoothing parameter (α)
Master iterationsSubproblem calls
0 0.2 0.4 0.6 0.8 0.95
102
103
smoothing parameter (α)
Time (sec.)
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Numerical Analysis
Sensivity to static α: Generalized Assignment
OR-Library C, D, E instances: 5 jobs per machine
0 0.2 0.4 0.6 0.8 0.95100
150
200
250
300
smoothing parameter (α)
Master iterationsSubproblem calls
0 0.2 0.4 0.6 0.8 0.950.5
1
1.5
2
2.5
smoothing parameter (α)
Time (sec.)
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Numerical Analysis
Sensivity to static α: Generalized Assignment
OR-Library C, D, E instances: 10 jobs per machine
0 0.2 0.4 0.6 0.8 0.95300
400
500
600
smoothing parameter (α)
Master iterationsSubproblem calls
0 0.2 0.4 0.6 0.8 0.95
40
60
80
100
smoothing parameter (α)
Time (sec.)
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Numerical Analysis
Sensivity to static α: Generalized Assignment
OR-Library C, D, E instances: 20 jobs per machine
0 0.2 0.4 0.6 0.8 0.95400
600
800
1,000
1,200
smoothing parameter (α)
Master iterationsSubproblem calls
0 0.2 0.4 0.6 0.8 0.95100
200
300
400
smoothing parameter (α)
Time (sec.)
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Numerical Analysis
Sensivity to static α: Generalized Assignment
OR-Library C, D, E instances: 40 jobs per machine
0.2 0.4 0.6 0.8 0.950
1,000
2,000
3,000
4,000
5,000
smoothing parameter (α)
Master iterationsSubproblem calls
0.2 0.4 0.6 0.8 0.950
0.5
1
1.5
2·104
smoothing parameter (α)
Time (sec.)
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Numerical Analysis
Sensivity to static α: Vertex Coloring
Number of calls to the oracle and Master CPU time
0 0.2 0.4 0.6 0.8 0.95
0.2
0.4
0.6
0.8
1
1.2
smoothing parameter (α)
num
bero
fsub
prob
lem
calls
queenmycielDSJ-1DSJ-2milesmugg
mulsolzeroinfpsol2school
average
0 0.2 0.4 0.6 0.8 0.950
0.2
0.4
0.6
0.8
1
1.2
smoothing parameter (α)
time
spen
tats
olvi
ngm
aste
rpro
blem
s queenmycielDSJ-1DSJ-2milesmugg
mulsolzeroinfpsol2school
average
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Numerical Analysis
Sensivity to static α: Vertex Coloring
Subproblem CPU time and Overall CPU time
0 0.2 0.4 0.6 0.8 0.95
0.5
1
1.5
2
smoothing parameter (α)
time
spen
tats
olvi
ngsu
bpro
blem
s queenmycielDSJ-1DSJ-2milesmugg
mulsolzeroinfpsol2school
average
0 0.2 0.4 0.6 0.8 0.95
0.5
1
1.5
smoothing parameter (α)
tota
lsol
ving
time
queenmycielDSJ-1DSJ-2milesmugg
mulsolzeroinfpsol2school
average
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Numerical Analysis
Sensivity to static α: Vertex Coloring
Dual price vector density & subproblem CPU timeon instance
queen12_12, using the oracle of [Ostergard, 2001]
0 100 200 300 400 5000
20
40
60
80
100
number of iterations
perc
enta
geof
nonz
ero
dual
s(%
)
0
0.2
0.4
0.6
0.8
1
·106
time
fors
olvi
ngsu
bpro
blem
nonzero(%)time
0 100 200 300 400 5000
20
40
60
80
100
number of iterationspe
rcen
tage
ofno
nzer
odu
als
(%)
0
0.2
0.4
0.6
0.8
1
·106
time
fors
olvi
ngsu
bpro
blem
nonzero(%)time
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
-
Numerical Analysis
Conclusions
Dual smoothing is the equivalent of IN-OUT separation↔
cross-fertilization with the cutting-plane community
Dual smoothing can have a large impact on
overallperformance.
Dual smoothing requires a single parameter; but it needs to
becarrefully set.
Hard instances for Kelley seems to be those with many
columns(f.i. more jobs per machine). Then, it seems best to apply
heavystabilization f.i. α= 0.85.
Dynamic and auto-adaptative α-schedules are to be derived.
Stabilization can make the pricing subproblem harder
andeliminates iterations where pricing is easy.
Colgen Workshop, June 2012 Col. Gen. Stabilization using Dual
Smoothing: Theory & Practice
Column Generation & Cut Separation in the DualStabilization
techniquesNumerical Analysis