Decomposition Methods

Lecture 6

Leonidas SakalauskasInstitute of Mathematics and InformaticsVilnius, Lithuania 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 the 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:

K

kkkk

T

zzzxyqpxc

k 1,...,,, 21

min

,, XxbAx

,kkkk hxTzW ,mk Rz Kk ,...,2,1

Block Diagonal

Staircase Systems

Block Angular

Benders Decomposition

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