Stability Analysis of Two-Stage Stochastic Mathematical Programs with Complementarity Constraints via NLP Regularization

Abstract

This paper presents numerical approximation schemes for a two-stage stochastic programming problem, where the second stage problem has a general nonlinear complementarity constraint. First the complementarity constraint is approximated by a parameterized system of inequalities with a well-known regularization approach by Scholtes [SIAM J. Optim., 11 (2001), pp. 918–936] in deterministic mathematical programs with equilibrium constraints; the distribution of the random variables of the regularized two-stage stochastic program is then approximated by a sequence of probability measures. By treating the approximation problems as a perturbation of the original (true) problem, we carry out a detailed stability analysis of the approximated problems, including continuity and local Lipschitz continuity of optimal value functions and outer semicontinuity and continuity of the set of optimal solutions and stationary points. A particular focus is given to the case where the probability distribution is approximated by the empirical probability measure which is also known as sample average approximation.