A Regularized Sample Average Approximation Method for Stochastic Mathematical Programs with Nonsmooth Equality Constraints

We investigate a class of two stage stochastic programs where the second stage problem is subject to nonsmooth equality constraints parameterized by the first stage variant and a random vector. We consider the case when the parametric equality constraints have more than one solution. A regularization method is proposed to deal with the multiple solution problem, and a sample average approximation method is proposed to solve the regularized problem. We then investigate the convergence of stationary points of the regularized sample average approximation programs as the sample size increases. The established results are applied to stochastic mathematical programs with $P_0$-variational inequality constraints. Preliminary numerical results are reported.