  Convex sample approximations of chance-constrained optimization problems are
considered, in which chance constraints are replaced by sets of sampled
constraints. We propose a randomized sample selection strategy that allows
tight bounds to be derived on the probability that the solution of the sample
approximation is feasible for the original chance constraints when a subset of
the sampled constraints is discarded. 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 approach that
meets required tolerances on violation probability given any specified a priori
and a posteriori probabilities. These bounds are tighter than confidence bounds
available for previously proposed repetitive scenario approaches, and we show
that the posterior bounds are exact for a particular problem subclass. The
approach is illustrated through numerical examples, and extensions to problems
involving multiple chance constraints are discussed.
