Necessary Optimality Conditions for Two-Stage Stochastic Programs with Equilibrium Constraints

Developing first order optimality conditions for two-stage stochastic mathematical programs with equilibrium constraints (SMPECs) whose second stage problem has multiple equilibria/solutions is a challenging undone work. In this paper we take this challenge by considering a general class of two-stage SMPECs whose equilibrium constraints are represented by a parametric variational inequality (where the first stage decision vector and a random vector are treated as parameters). We use the sensitivity analysis on deterministic mathematical programs with equilibrium constraints (MPECs) as a tool to deal with the challenge: First, we extend a well-known theorem in nonsmooth analysis about the exchange of the subdifferential operator with Aumann’s integration from a nonatomic probability space to a general setting; second, we apply the extended result together with the existing sensitivity analysis results on the value function of the deterministic MPEC and the bilevel programming to the value function of our second stage problem; third, we develop various optimality conditions in terms of the subdifferential of the value function of the second stage problem and its relaxations which are constructed through the gradients of the underlying function at the second stage; finally we analyze special cases when the variational inequality constraint reduces to a complementarity problem and further to a system of nonlinear equalities and inequalities. The subdifferential to be used in this paper is the limiting (Mordukhovich) subdifferential, and the probability space is not necessarily nonatomic which means that Aumann’s integral of the limiting subdifferential of a random function may be strictly smaller than that of Clarke’s.