Sample Average Approximation Method for Chance Constrained ...

y a version of the uniform law of large numbers (see, [9, Proposition 7, p.363]).By lower semicontinuity of p(x) and ˆp N (x) we have that the feasible sets of the‘true’ problem (4) and its SAA counterpart (5) are closed sets. Therefore, if the setX is bounded (i.e., compact), then problems (4) and (5) have nonempty sets of optimalsolutions denoted, respectively, as S and ŜN, provided that these problems havenonempty feasible sets. We also denote by ϑ ∗ and ˆϑ N the optimal values of the trueand the SAA problems, respectively. The following result shows that for γ = α, undermild regularity conditions, ˆϑ N and ŜN converge w.p.1 to their counterparts of the trueproblem.We make the following assumption.(A) There is an optimal solution ¯x of the true problem (4) such that for any ε > 0there is x ∈ X with ‖x − ¯x‖ ≤ ε and p(x) < α.In other words the above condition (A) assumes existence of a sequence {x k } ⊂ Xconverging to an optimal solution ¯x ∈ S such that p(x k ) < α for all k, i.e., ¯x is anaccumulation point of the set {x ∈ X : p(x) < α}.Proposition 2.2 Suppose that the significance levels of the true and SAA problems arethe same, i.e., γ = α, the set X is compact, the function f(x) is continuous, G(x, ξ) is10