col. gen. stabilization using dual smoothing: theory ...€¦ · outline 1 column generation & cut...

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


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}



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)



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)



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}



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)



Dual Polyhedra: Outer and Inner approximations

π

η

L(π)

(πt ,ηt)(π̂, L̂)

π

η

L(π)



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]

...



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


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



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).



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



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



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



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



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



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


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



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.



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


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


Time (sec.)


Numerical Analysis


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



0 0.2 0.4 0.6 0.8 0.95

4

6

8

10

12


Time (sec.)


Numerical Analysis


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



0 0.2 0.4 0.6 0.8 0.950

50

100

150

200


Time (sec.)


Numerical Analysis


100 jobs per machines100 jobs / 1 machine

0 0.2 0.4 0.6 0.8 0.95

103

104



0 0.2 0.4 0.6 0.8 0.95

102

103


Time (sec.)


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



0 0.2 0.4 0.6 0.8 0.950.5

1

1.5

2

2.5


Time (sec.)


Numerical Analysis



0 0.2 0.4 0.6 0.8 0.95300

400

500

600



0 0.2 0.4 0.6 0.8 0.95

40

60

80

100


Time (sec.)


Numerical Analysis



0 0.2 0.4 0.6 0.8 0.95400

600

800

1,000

1,200



0 0.2 0.4 0.6 0.8 0.95100

200

300

400


Time (sec.)


Numerical Analysis



0.2 0.4 0.6 0.8 0.950

1,000

2,000

3,000

4,000

5,000



0.2 0.4 0.6 0.8 0.950

0.5

1

1.5

2·104


Time (sec.)


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


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


time

spen

tats

olvi

ngm

aste

rpro

blem

s queenmycielDSJ-1DSJ-2milesmugg


average


Numerical Analysis


Subproblem CPU time and Overall CPU time

0 0.2 0.4 0.6 0.8 0.95

0.5

1

1.5

2


time

spen

tats

olvi

ngsu

bpro

blem

s queenmycielDSJ-1DSJ-2milesmugg


average

0 0.2 0.4 0.6 0.8 0.95

0.5

1

1.5


tota

lsol

ving

time

queenmycielDSJ-1DSJ-2milesmugg


average


Numerical Analysis


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


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.


Column Generation & Cut Separation in the DualStabilization techniquesNumerical Analysis

col. gen. stabilization using dual smoothing: theory ...€¦ · outline 1 column generation & cut...

Documents