General Properties of Two-Stage Stochastic Programming Problems with Probabilistic Criteria

Two-stage stochastic programming problems with the probabilistic and quantile criteria in the general statement are considered. Sufficient conditions for the measurability of the loss function and also for the semicontinuity of the criterion functions are given. Sufficient conditions for the existence of optimal strategies are established. The equivalence of the a priori and a posteriori statements of the problems under study is proved. The application of the confidence method, which consists in the transition to a deterministic minimax problem, is described and justified. Sample approximations of the problems are constructed and also conditions under which the optimal strategies in the approximating problems converge to the optimal strategy in the original problem are presented. The results are illustrated by an example of the linear two-step problem. The two-stage problem with the probabilistic criterion is reduced to a mixed-integer problem.