Research On Several Theories And Applications Of Mathematical Programming With Equilibrium Constraints

Mathematical programming with equilibrium constraints is an arousing general interests problem that recently developed in mathematical programming field, it can be thought as a bilevel programming with variational inequalities or complementarity constraints. Variational inequalities or complementarity constraints can well depict many equilibrium problems in engineering and economy, however, the abstract constraints also make this problem become difficult to deal with. So, how to transform this problem and present some easier equivalent forms, and to study the optimality conditions and algorithms become very important.This dissertation first systematically and comprehensively generalizes current status quo study of mathematical programming with equilibrium constraints throughout the world, and then analyzes the existing difficulties and complexities of this problem. By studing the Nash Game and variational inequalities, Nash Game and Stackelberg Game, as well as the general bilevel programming and mathematical programming with equilibrium constraints, we point out the corresponding relationships among them. Second, basing on the common used equivalent forms, we further extend the existing results. The equivalent formulations are respectively presented in generalized complementary constraint and K-T condition constraint and fixed point constraint, where C (x)is a closed convex set represented by finitely many inequalities and equalities or a general closed convex set. At last, basing on the comprehensive study of exact penalty function theories of mathematical programming with equilibrium constraints, and according to the long time existing questions of price control problem, we establish two equivalent exact penalty forms of it with the help of the related theories of mathematical programming with equilibrium constraints problem and others, the validity of our results are confirmed by a open numerical example.The results abstained in this dissertation make an improvement of the theory of mathematical programming, and offer some basis for the study of algorithms and some potential applications of this kind problem in the future.