Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization

In this paper, we consider the stochastic mathematical programs with equilibrium constraints, which includes two kinds of models called here-and-now and lower-level wait-andsee problems. We present a combined smoothing implicit programming and penalty method for the problems with a finite sample space. Then, we suggest a quasi-Monte Carlo approximation method for solving a problem with continuous random variables. A comprehensive convergence theory is included as well. We further report numerical results with the so-called picnic vender decision problem.