Stochastic Programming - Index of

166 STOCHASTIC PROGRAMMING
Figure 5
Generation of feasibility cuts.
to our assumption. We also see that if t gets large enough, the problem is
always feasible. This is what we solve for all ξ ∈A.Ifforsomeξ we find a
positive optimal value, we have found a ξ for which h(ξ) − T (ξ)ˆx ∉ pos W ,
and we create the cut
σ T (h(ξ) − T (ξ)x) ≤ 0 ⇐⇒ σ T T (ξ)x ≥ σ T h(ξ). (2.1)
The σ used here is a generator of pol pos W , but it is not in general as close
to h(ξ) − T (ξ)ˆx as possible. This is in contrast to what would have happened
hadweusedthel 2 norm. (See Example 3.1 below for an illustration of this
point.)
Note that if T (ξ) ≡ T 0 , the expression σ T T 0 x in (2.1) does not depend on
ξ. Since at the same time (2.1) must be true for all ξ, we can for this special
case strengthen the inequality by calculating
σ T T 0 x ≥ σ T (
h 0 +max σ T H ) t.
t∈Ξ
Since σ T T 0 is a vector and the right-hand side is a scalar, this can conveniently
be written as −γ T x ≥ δ. Theˆx we started out with will not satisfy this
constraint.
Example 3.1 We present this little example to indicate why the l 1 and l 2
norms give different results when we generate feasibility cuts. The important
point is how the two norms limit the possible σ values. The l 1 norm is given
in the left part of Figure 6, the l 2 norm in the right part.