Decomposition Methods

1. Decomposition Methods Lecture 6 Leonidas Sakalauskas Institute of Mathematics and Informatics Vilnius, Lithuania EURO Working Group on Continuous Optimization

2. Content • Constraint matrix block systems • Benders decomposition • Master problem and cuts • Dantzig-Wolfe decomposition • Comparison of Benders and Dantzig-Wolfe decompositions

3. Two-stage SLP The two-stage stochastic linear programming problem can be stated as

4. Two-Stage SLP Assume the set of scenarios K be finite and defibed by probabilities In continuous stochastic programming by the Monte-Carlo Method this is equivalent to

5. Two-Stage SLP Using the definition of discrete random variable the SLP considered is equivalent to large linear problem with block constraint matrix:

6. Block Diagonal

7. Staircase Systems

8. Block Angular

9. Benders Decomposition

20. Feasibility

23. Dantzif-Wolfe Decomposition Primal Block Angular Structure

24. The Problem

39. Wrap-Up and conclusions • The discrete SLP is reduced to equivalent linear program with block constraint matrix, that solved by Benders or Dantzig-Wolfe decomposition method • The continuous SLP is solved by decomposition method simulating the finite set of random scenarios