Lecture 6
Decomposition Methods
Leonidas Sakalauskas
Institute of Mathematics and Informatics
Vilnius, Lithuania
EURO Working Group on Continuous Optimization
Content
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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
F ( x)  c  x  Emin y q  y  min
W  y  T  x  h,
Ax  b,
y  Rm ,
x X.
Two-Stage SLP
Assume the set of scenarios K be finite and
defibed by probabilities
p1 , p2 ,..., pK ,
In continuous stochastic programming by
the Monte-Carlo Method this is equivalent
to
1
pi 
N
Two-Stage SLP
Using the definition of discrete random variable
the SLP considered is equivalent to large linear
problem with block constraint matrix:
K
min
x , z1 , z 2 ,..., z k
c  x   pk  qk  y k
T
k 1
m
z

R
 ,
Wk  zk  Tk  x  hk , k
Ax  b,
x X,
k  1,2,..., K
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