Department of Applied Mathematics & Physics, Kyoto University

### Technical Report 2003-009 (July 16, 2003) Approximation and Convergence in Stochastic Linear Complementarity Problems by Xiaojun Chen and Masao Fukushima

The stochastic linear complementarity problem is formulated as a problem of minimizing an expected residual defined by an NCP function. We generate observations by the quasi-Monte Carlo methods and show that every accumulation point of minimizers of discrete approximation problems is a minimum residual solution of the stochastic linear complementarity problem. We show that a sufficient condition for the existence of a solution to the residual minimization problem and its discrete approximations is that there is an observation $\omega^i$ such that the cost matrix $M(\omega^i)$ is an $R_0$ matrix. Furthermore, we show that a class of fixed cost problems are continuously differentiable and can be solved without using discrete approximation.