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# Decomposition Methods in SLP

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AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 6.

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### Decomposition Methods in SLP

1. 1. Lecture 6 Decomposition Methods in SLP Leonidas Sakalauskas Institute of Mathematics and Informatics Vilnius, Lithuania <sakal@ktl.mii.lt> EURO Working Group on Continuous Optimization
2. 2. Content  Constraint matrix block systems  Benders decomposition  Master problem and cuts  Dantzig-Wolfe decomposition  Comparison of Benders and Dantzig-Wolfe decompositions
3. 3. Two-stage SLP The two-stage stochastic linear programming problem can be stated as F ( x) c x E min y q y min W y T x h, y Rm , Ax b, x X.
4. 4. Two-Stage SLP Assume the set of scenarios K be finite and defibed by probabilities p1 , p2 ,..., pK , In continuous stochastic programming by Monte-Carlo method this is equivalent to 1 pi N
5. 5. Two-Stage SLP Using the definition of discrete random variable the SLP considered is equivalent to large linear problem with block constraint matrix: q T T min c x pi q yi x , y1 , y2 ,..., yq i 1 Wi yi Ti x hi , yi Rp, i 1,2,..., q Ax b, x X,
6. 6. Block Diagonal
7. 7. Staircase Systems
8. 8. Block Angular
9. 9. Benders Decomposition
10. 10. P: F ( x) c x q y min W y T x h, Ax b x Rn , y Rm , F ( x) c x z( x) min Ax b x Rn , z ( x) min q y y W y T x h, y Rm ,
11. 11. Primal subproblem qT y min y m W y h T x y R , Dual subproblem u T (h T x) max u u WT q 0
12. 12. Feasibility
13. 13. Dantzif-Wolfe Decomposition Primal Block Angular Structure
14. 14. The Problem
15. 15. 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