Stochastic Programming - Index of

244 STOCHASTIC PROGRAMMING
Proposition 4.1 The feasible set
is convex.
B(1) := {x | P ({ξ | T (ξ)x ≥ h(ξ)}) ≥ 1}
Proof Assume that x, y ∈B(1) and that λ ∈ (0, 1). Then for Ξ x := {ξ |
T (ξ)x ≥ h(ξ)} and Ξ y := {ξ | T (ξ)y ≥ h(ξ)} we have P (Ξ x )=P (Ξ y )=1.
As is easily shown, this implies for Ξ ∩ := Ξ x ∩ Ξ y that P (Ξ ∩ ) = 1.
Obviously, for z := λx +(1− λ)y we have T (ξ)z ≥ h(ξ) ∀ξ ∈ Ξ ∩ such that
{ξ | T (ξ)z ≥ h(ξ)} ⊃Ξ ∩ . Hence we have z ∈B(1). ✷
Considering once again the example illustrated in Figure 18 in Section 1.6,
we observe that if we had required a reliability α>93%, the feasible set
would have been convex. This is a consequence of Proposition 4.1 for discrete
distributions, and may be stated as follows.
Proposition 4.2 Let ˜ξ have a finite discrete distribution described by P (ξ =
ξ j )=p j , j =1, ···,r (p j > 0 ∀j). Then for α>1 − min j∈{1,···,r} p j the
feasible set
B(α) :={x | P ({ξ | T (ξ)x ≥ h(ξ)}) ≥ α}
is convex.
Proof: The assumption on α implies that B(α) =B(1) (see Exercises at the
end of this chapter).
✷
In conclusion, for discrete distributions and reliability levels chosen “high
enough” we have a convex problem. Replacing E˜ξc(˜ξ) byc, we then simply
have to solve the linear program (provided that X is convex polyhedral)
min x∈X c T x
s.t. T (ξ j )x ≥ h(ξ j ), j =1, ···,r.
This observation may be helpful for some particular chance-constrained
problems with discrete distributions. However, it also tells us that for chanceconstrained
problems stated with continuous-type distributions and requiring
a reliability level α