The Sample Average Approximation Method For Solving Stochastic Complementarity Problems And Its Convergence Analysis

Stochastic complementarity problems are an important part in minimization prob-lems. It has many applications in several areas, for example:the traffic equilibrium prob-lem with stochastic demands, the problem of market demand with demand uncertainty, the control problem with stochastic turbulence. Because there are more applications can be seen in actual problems, thus stochastic complementarity problems become the focus of attention. In this paper, we study two types of stochastic complementarity problems:One is stochastic nonlinear complementarity problems and the other is stochastic generalized second-order cone complementarity problems. For stochastic nonlinear complementarity problems, basing on the theory of CVaR, we use restricted nonlinear complementarity function (NCP function) to build the loss function in portfolio optimization and present conditional value-at-risk (CVaR) model for solving stochastic nonlinear complementarity problems. Since the CVaR model contains an expectation function and a nonsmooth function, we then employ sample average approximation method and smoothing method to give approximation problems of CVaR model. Furthermore, an algorithm is given for solving these approximation problems. On theory, the boundedness of level set of CVaR model and the convergence results of global solutions sequence and stationary points sequence for the corresponding approximation problems are considered. Above results ensure that the proposed new model and the corresponding approximation problems are feasible for solving stochastic nonlinear complementarity problems on theory. In addition, the simple numerical experiments show that the above approach is effective. For stochas-tic generalized second-order cone complementarity problems, we use the merit functions to reformulate the stochastic generalized of a second-order cone complementarity prob-lems as box-constrained optimization problems.Since the box-constrained minimization problems contain an expectation function, we then employ sample average approximation method to give approximation problems of the box-constrained minimization. Meanwhile the convergence results of global solutions sequence and stationary points sequence for the corresponding approximation problems are considered.