The ERM Model For Solving Box Constrained Stochastic Vector Variational Inequality Problems And Convergence Analysis Of Their Approximation Problems

The vector variational inequality problem is a generalization of the classical variational inequality problem,and it is also a kind of equilibrium problem in the field of optimization,which has wide applications in economics,management,engineering and other fields.At present,some results have been achieved in the research on vector variational inequality problems,and the theoretical research on this problem has gradually formed a perfect system.However,in practical applications,the decision-maker‘s environment faced by decision makers is often affected by a large number of uncertain factors,such as environmental changes and sudden accidents.If the influence of these uncertainties is not considered when solving the problem,the decision made may have a large error,resulting in a wrong decision.Therefore,it is of great practical significance to study the stochastic vector variational inequality problem with random factors.The box-constrained stochastic vector variational inequality problem considered in this paper is a special case of stochastic vector variational inequality problem,which also plays an irreplaceable role in the practical application of economic management and many other aspects.For the box-constrained stochastic vector variational inequality problem,in general,there is no solution such that the value of almost all random variables is true.However,in order to meet the needs of this problem in practical application,and based on the research methods and research results of scholars about the problem of stochastic variational inequality and stochastic complementarity,this paper gives a deterministic model for solving the problem of box-constrained stochastic vector variational inequality,that is,the expected residual minimization model,and takes the solution of this model as the solution to the box-constrained stochastic vector variational inequality problem.In this paper,a gap function equivalent to the box-constrained stochastic vector variational inequality problem is first given,and based on this gap function,a ERM model for solving the box-constrained stochastic vector variational inequality problem is proposed.Since the proposed expected residual minimization model contains mathematical expectation calculation,this calculation is generally not easy to obtain,so this paper will use the quasi-Monte Carlo method to solve the sample average approximation problem of the expected residual minimization model.Furthermore,this paper theoretically proves that when the sample space is a compact set,the global optimal solution sequence of the proposed sample average approximation problem converges to the global optimal solution of the original expected residual minimization model.The theoretical results obtained can ensure that it is reasonable to use the solution of the proposed sample mean approximation problem as the solution to the original expected residual minimization problem under certain error bound.In addition,when the sample space does not meet the conditions of the compact set,this paper proposes the compact approximation problem corresponding to the expected residual minimization model,and theoretically proves the convergence results of the sequence of the global optimal solution of the compact approximation problem and the global optimal solution of the expected residual minimization model.