Monte Carlo and quasi-Monte Carlo sampling methods for a class of stochastic mathematical programs with equilibrium constraints

Abstract

In this paper, we consider a class of stochastic mathematical programs with equilibrium constraints introduced by Birbil et al. (Math Oper Res 31:739–760, 2006). Firstly, by means of a Monte Carlo method, we obtain a nonsmooth discrete approximation of the original problem. Then, we propose a smoothing method together with a penalty technique to get a standard nonlinear programming problem. Some convergence results are established. Moreover, since quasi-Monte Carlo methods are generally faster than Monte Carlo methods, we discuss a quasi-Monte Carlo sampling approach as well. Furthermore, we give an example in economics to illustrate the model and show some numerical results with this example.