AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 6. More info at http://summerschool.ssa.org.ua

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. 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
F ( x) c x E min y q y min
W y T x h, y Rm ,
Ax b, x X.

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. 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,

7. Block Diagonal

8. Staircase Systems

9. Block Angular

10. Benders Decomposition

11. 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 ,

12. 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

19. Feasibility

20. Dantzif-Wolfe Decomposition
Primal Block Angular Structure

21. The Problem

36. 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