Several Theoretical Problems Of Variational Inequalities And Optimization Problems With Equilibrium Constraints

The dissertation studies the sovability of variational inequalities and feasibility and optimality conditions for optimization problems with equilibrium constraints or mathematical programs with equilibrium constraints(MPEC for short), which is organized as follows:First, we use a new method based on the concept of exceptional family to study the sovability of variational inequalities. A new concept of exceptional family is introduced to study the solvability of variational inequalities in the Euclidean space Rn. We prove that if there does not exist an exceptional family of elements for continuous function /, then the corresponding variational inequality has a solution. A related existence theorem for the solution to the variational inequality is proved. Then, we extend the concept of exceptional family in Rn to general Hilbert space, prove the corresponding basic theorem and related existence theorems for the solution to the variational inequality. Moreover, we generalize the concept of exceptional family of variational inequality to quasi-variational inequality in Rn and apply it to study the existence of the solution to quasi-variational inequality.Second, we study variational inequalities with nonmonotone mapping. We introduce a new concept of relaxed η- a pseudomonotonicity as well as a class of variational-like inequalities with relaxed η - a pseudomonotone mappings. Using the KKM technique, we obtain the existence of solutions for variational-like inequalities with relaxed η - a pseudomonotone mappings in a reflexive Banach space. We also study the existence of solutions for variational inequalities with relaxed densely u-pseudomonotone in normed linear spaces. Some particular cases in reflexive Banach spaces are presented which include several previously known results.Finally, we study feasibility and optimality conditions for optimization problems with equilibrium constraints. Two feasibility conditions in MPECs is introduced. We show that the two feasibility conditions are different from that of Fukushima and Pang and that of Wan, and derive first-order necessary optimality conditions for MPECs. Then we derive first-order necessary optimality conditions for optimization problems with affine variational inequality, which joint upper-level feasible region Z includes a linear inequality constraint and a linear equality constraint. Moreover, A very general optimization problems with a complemen-tarity problem constraint, a inequality and a equality constraints, and an abstract constraint are studied. Fritz John type necessary optimality conditions involving Mordukhovich coderivatives are derived.