Sample Average Approximation Methods For Solving Stochastic Mixed Complementarity Problems

In this paper, we discuss two kinds of sample average approximation method for stochastic mixed complementarity problems (SMCP):1. Expected value (EV) method:We first present an EV model for SMCP and use the so-called Fischer-Burmeister function to reformulate the EV model as nonsmooth equa-tions. Then, we employ the sample average approximation techniques, which are based on Monte Carlo method, to propose a semismooth Newton method for solving the equa-tions. Convergence analysis is given. We finally apply these results to traffic equilibrium problems and give some preliminary numerical results.2. Expected residual minimization (ERM) method:We first construct an ERM model for SMCP by using the penalized Fischer-Burmeister function and consider some prop-erties of the ERM formulation, including boundedness of level sets and error bounds. Then we employ the sample average approximation techniques to approximate the prob-lem. Convergence analysis is also established. Preliminary numerical experiments indicate that the proposed method is applicable.