This paper considers convex sample approximations of chance-constrained optimization problems, in which the chance constraints are replaced by sets of sampled constraints. We show that, if a subset of sampled constraints are discarded, then the use of a randomized sample selection strategy allows tight bounds to be derived on the probability that the solution of the sample approximation is feasible for the original chance constraints. These confidence bounds are shown to be tighter than the bounds that apply if constraints are discarded according to optimal or greedy discarding strategies. We further show that the same confidence bounds apply to solutions that are obtained from a two stage process in which a sample approximation of a chance-constrained problem is solved, then an empirical measure of the violation probability of the solution is obtained by counting the number of violations of an additional set of sampled constraints. We use this result to design a repetitive scenario ap 查看全文>>