Decomposition Methods in SLP

Lecture 6

Leonidas SakalauskasInstitute of Mathematics and InformaticsVilnius, Lithuania <sakal@ktl.mii.lt>

EURO Working Group on Continuous Optimization

Content

Constraint matrix block systems

Benders decomposition

Master problem and cuts

Dantzig-Wolfe decomposition

Comparison of Benders and Dantzig-Wolfe decompositions

Two-stage SLP

The two-stage stochastic linear programming problem can be stated as

minmin)( yqExcxF y

,hxTyW,mRy

., XxbAx

Two-Stage SLP

Assume the set of scenarios K be finite and defibed by probabilities

,,...,, 21 Kppp

In continuous stochastic programming by Monte-Carlo method this is equivalent to

Npi

1

Two-Stage SLP

Using the definition of discrete random variable the SLP considered is equivalent to large linear problem with block constraint matrix:

q

i

i

T

i

T

yyyxyqpxc

q 1,...,,, 21

min

,, XxbAx

,iiii hxTyW ,p

i Ry qi ,...,2,1

Block Diagonal

Staircase Systems

Block Angular

Benders Decomposition

min)( yqxcxF

,hxTyW

,mRy

,nRx

bAx

min)()( xzxcxF

bAx

,nRx

yqxzy

min)(

,hxTyW ,mRy

P:

Primal subproblem

y

T yq min

xThyW ,mRy

Dual subproblem

u

T xThu max)(

0qWu T

Feasibility

Dantzif-Wolfe Decomposition Primal Block Angular Structure

The Problem

Wrap-Up and conclusions

oThe discrete SLP is reduced to equivalent linear program with block constraint matrix, that solved by Benders or Dantzig-Wolfe decomposition method

o The continuous SLP is solved by decomposition method simulating the finite set of random scenarios