Research On Sample Average Approximation Method For Stochastic Variational Inequality Problems

In the middle1960s, people start to study the variational inequality problem. As time goes on, the theory and the applications of this problem has got good achievements. In the theory, the achievements include:a series of gap functions are proposed, and their properties are better than ever; In the applications, it has been used in engineering, economics and so on. So, with the development of the actual situation, on the basis of the variational inequality problem, people begin to study it with random variable. Because of the uncertain factor, the variational inequality problem is more complex and difficult to calculate directly. In view of this situation, scholars have constructed two reasonable certainty models(EV model and ERM model) to solve it. And in the process of solving this problem, people prefer to use the regulared gap function.For the EV model of the stochastic variational inequality problems, this paper presents a method to solve them, and gives the detailed numerical results. The main results obtained from this paper are as follows:1. This paper gives two properties of the D-gap function and the processes of the proof.2. Based on the D-gap function, this paper gives the equal unconstrained and constrained optimization model of the EV model of the stochastic variational inequality problems.3. By applying the sample average approximation method, this paper gets the sample average approximation problem of the constrained optimization problem, and proves objective functions of the sample average approximation problem uniformly convergence and epi-convergence on target function of the initial optimization problem under some conditions. This paper further proves that, the optimal values and the optimal solutions of the sample average approximation problems convergent to the optimal value and the optimal solution of the initial optimization problem with probability1under certain conditions.4. In order to prove the efficiency of this method, this paper conducts numerical experiments on four examples with Matlab programming. In order to find the global optimal solution, this paper combines with the particle swarm algorithm to get the numerical results. By comparing the numerical results of the regular-gap function and the D-gap function, this paper obtains the conclusion:the D-gap function is more effective than the regularized gap function.