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Decomposition Methods in SLP
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Decomposition Methods in SLP
Lecture 6
Leonidas SakalauskasInstitute of Mathematics and InformaticsVilnius, Lithuania <[email protected]>
EURO Working Group on Continuous Optimization
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Content
Constraint matrix block systems
Benders decomposition
Master problem and cuts
Dantzig-Wolfe decomposition
Comparison of Benders and Dantzig-Wolfe decompositions
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Two-stage SLP
The two-stage stochastic linear programming problem can be stated as
minmin)( yqExcxF y
,hxTyW,mRy
., XxbAx
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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
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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
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Block Diagonal
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Staircase Systems
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Block Angular
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Benders Decomposition
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min)( yqxcxF
,hxTyW
,mRy
,nRx
bAx
min)()( xzxcxF
bAx
,nRx
yqxzy
min)(
,hxTyW ,mRy
P:
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Primal subproblem
y
T yq min
xThyW ,mRy
Dual subproblem
u
T xThu max)(
0qWu T
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Feasibility
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Dantzif-Wolfe Decomposition Primal Block Angular Structure
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The Problem
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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