decomposition methods
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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
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