Quantitative stability analysis of stochastic mathematical programs with vertical complementarity constraints

Abstract

This paper studies the quantitative stability of stochastic mathematical programs with vertical complementarity constraints (SMPVCC) with respect to the perturbation of the underlying probability distribution. We first show under moderate conditions that the optimal solution set-mapping is outer semiconitnuous and optimal value function is Lipschitz continuous with respect to the probability distribution. We then move on to investigate the outer semiconitnuous of the M-stationary points by employing the reformulation of stationary points and some stability results on the stochastic generalized equations. The particular focus is given to discrete approximation of probability distributions, where both cases that the sample is chosen in a fixed procedure and random procedure are considered. The technical results lay a theoretical foundation for approximation schemes to be applied to solve SMPVCC.