Stochastic Programming - Index of

172 STOCHASTIC PROGRAMMING
θ
Q(x)
cx+θ
x
cut 5
0
x 3 x 1
x 2 cut 3 cut 1
cut 2
( x 5 ,θ 5
)
cut 4
( x 4 , θ 4
)
Figure 11
Example of the progress of the L-shaped decomposition algorithm.
Since there are finitely many feasible bases coming from the matrix W ,there
are only finitely many such cuts.
We are now ready to present the basic setting of the L-shaped decomposition
algorithm. It is shown in Figure 10. To use it, we shall need a procedure
that solves LPs. It can be found in Figure 7. Also, to avoid too complicated
expressions, we shall define a special procedure for solving the master problem;
seeFigure8.Furthermore,werefertoprocedure pickξ(A,ξ), which simply
picks an element ξ from the set A, and, finally, we use procedure feascut
which is given in Figure 9. The set A was defined on page 162.
In the algorithms to follow, let −Γx ≥ ∆ represent the K feasibility
cuts −γ T k x ≥ δ k,andlet−βx + Iθ ≥ α represent the L optimality cuts
−β T l x + θ ≥ α l. Furthermore, let e be a column of 1s of appropriate size.