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Decomposition Methods. Lecture 6. Leonidas Sakalauskas Institute of Mathematics and Informatics Vilnius, Lithuania EURO Working Group on Continuous Optimization. Content. Constraint matrix block systems Benders decomposition Master problem and cuts Dantzig-Wolfe decomposition - PowerPoint PPT Presentation
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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