Regularized Sample Average Approximation Method For Solving Stochastic Linear Complementarity Problems

Stochastic variational inequality and complementarity problems, as important top-ics of mathematical programming theory, have many applications in many fields such as engineering design, optimal control, information technology and economic equilibrium. The sample average approximation method is one of effective approaches for solving the stochastic complementarity problems by using Monte Carlo methods.On the basis of previous work, we study a regularized sample average approximation (SAA) method to solve the stochastic linear complementarity problems (SLCP). Firstly, we discribe the regularized approximation method for SLCP by using a penalized Fischer-Burmeister function, and prove that any accumulation point of stationary points of the corresponding regularized approximation problems is a stationary point of the expected residual minimization (ERM) problem. Then, we propose a regularized SAA approach, which is based on the Monte Carlo method, to give approximation problems for the ERM problem, and show that every accumulation point of the global optimal solutions (or sta-tionary points) of the corresponding regularized SAA problems is optimal (or stationary) to the ERM problem under mild conditions. Next, we give a procedure to generate a test problem of monotone SLCP. Preliminary numerical experiments on the test problems in-dicate that the proposed approach is effective and feasible. Finally, we apply the obtain results to a realistic application, the market equilibrium under uncertainty.