The Sample Average Approximation Method For The Stochastic Variational Inequality Problem

Variational Inequality Problems(In short VIPs) firstly appeared in the Physics equations. Lions J L and Tampacchia G S first established the initial theory of Variational Inequality in 1964.With the great improvement in the theory of variational inequalities,VIPs have been effectively applied to modeling a range of equilibrium problems in engineering,economics, transportation,operations research,and networks.As many practical problems may involve uncertain factors,by modeling,models obtained are variational inequalities models where uncertainties are involved.We call them stochastic variational inequalities.In the literature, many researers and practicers have begun to have increasing interesting in stochastic variational inequality problems,but there only exist a few papers on stochastic variational inequality problems.The sample average approximation method is one of effective approachs for solving stochastic optimization problems by using Monte Carlo simulation.In this technique the expected objective function of the stochastic problem is approximated by a sample average function.Further,the sample average problem is solved by deterministic optimization methods. The solution of the sample average problem is a reasonable approximation solution of the original problems.On the basis of previous work,we make use of the sample average approximation method to find solutions of stochastic variational inequality problems.In this thesis,I will formulate the staochastic variational inequality problem as simply constrained optimization problem by using regularized gap function.Based on this reformulation,I proposed sample-average approximation method for solving this stochastic variational inequality problem and prove its convergence.In order to verify effectiveness and feasibility of the proposed Sample average approximation method,A few simple testings of stochastic variational inequality problems are given.Numerical results show our method is very effective and feasible.