Constraint Qualifications For Optimization Problem With Variational Inequality Constraints And Applications

The optimization problem with variational inequality constraints(OPVIC)is an important model in the field of operations research,which includes the mathematical programming with complementary constraints as its special case and has important applications in many fields,such as finance,computer science,transportation,engineering science,machinery,electrical engineering and so on.However,it cannot be considered as a general constrained optimization problem,because some constraint qualifications for general constrained optimization problems are not valid for this problem.The constraint qualifications of OPVIC have been studied by many scholars.However,the relationships between them and the stability of feasible set and the expression of the normal cone have not been completely established.The study of such theoretical problems can lay the foundation for the study of the stability and optimality conditions of OPVIC.Therefore,it is of great theoretical value and practical significance to study the constraint qualifications for OPVIC and their applications.In this paper,we establish the relationships between the constraint qualification of OPVIC and the perturbed solution mapping of the constrained system and the stability of the full perturbation solution mapping,and obtain sufficient conditions for the expressions of normal cone and regular normal cone of feasible region of OPVIC at certain points.The concrete content of this paper are as follows:Firstly,the various constraint qualifications of OPVIC are summarized and the relationships between them are discussed,and the relationships between the constraint qualifications of OPVIC and stability of the perturbation solution mapping and the full perturbation solution mapping of the constrained system are established.In particular,the relationships between the no nonzero normal multiplier constraint qualification and the Aubin property of the perturbation solution mapping and the full perturbation solution mapping of the constrained system have been established.Secondly,the relationships between the stability of the perturbation solution mapping and full perturbation solution mapping of the constrained system and the normal cone of the feasible region of OPVIC are established.The sufficient conditions for expressions of the normal cone and the regular normal cone of feasible region of OPVIC at certain points are obtained,that is,the stability of the perturbation solution mapping of the constrained system can ensure the expression of regular normal cone of feasible region of OPVIC at a localoptimal solution,and ensure the specific expression of the normal cone under a additional constraint qualification.Finally,we apply the constraint qualifications of OPVIC to the stochastic programming problem.Under certain constraint qualifications,it is proved that when the number of samples tends to infinity,the sequence of optimal solutions of sample average approximation problems of stochastic programming with variational inequality constraints(SPVIC)tends to the true problem with probability one,which provides a theoretical support for solving the stochastic programming with variational inequality constraints by sample average approximation method.