Some Studies On Optimization Problems With Equilibrium Constraints

Mathematical program with equilibrium constraints (MPEC) plays a very important role in many fields such as engineering design, economic equilibria, transportation science, multilevel game, and mathematical programming itself. Recently, as further application-s of MPECs, stochastic mathematical programs with equilibrium constraints (SMPEC) have attracted people’s attention. On the other hand, deterministic multiobjective prob-lems with equilibrium constraints (MOPEC) have been studied. But, little researchers consider the stochastic multiobjective optimization problem with equilibrium constraints which has many application in practice. Problems with second-order cone (SOC) con-straints also attract much attention of many researchers. However, little study has been done on the mathematical program with second-order cone complementarity constrains.The main results of this dissertation can be summarized as follows:1. In Chapter2, we consider the mathematical program with vertical complementari-ty constraints. We show that the min-max-min problems and the problems with max-min constraints can be reformulated as the above problem. As a complement of the work of Scheel and Scholtes in2000, we derive the Mordukhovich-type stationarity conditions for the considered problem. We further reformulate various popular stationarity systems as nonlinear equations with simple constraints. A modified Levenberg-Marquardt method is employed to solve these constrained equations.2. In Chapter3, we consider the mathematical programs with vertical complementar-ity constraints (MPVCC). We present a relaxed program for this problem. We show that the linear independence constraint qualification holds for the relaxed problem under some mild conditions. We also consider a limiting behavior of the relaxed problem. We prove that any accumulation point of stationary points of the relaxed problems is C-stationary to the original problem under the MPVCC linear independence constraint qualification and they are M-stationary if, in addition, an approaching subsequence satisfies second order necessary condition.3. In Chapter4, we consider a class of stochastic multiobjective problems with complementarity constraints (SMOPCC). We derive the first-order optimality conditions including the Clarke/Mordukhovich/strong-type stationarity in the Pareto sense for the SMOPCC. Since these first-order optimality systems involve some unknown index sets, we reformulate them as nonlinear equations with simple constraints. Then, we introduce an asymptotic method to solve these constrained equations. Furthermore, we apply this methodology results to a patient allocation problem in healthcare management.4. In Chapter5, we considers the mathematical program with second-order cone complementarity constrains (MPSOCC). As a generalization of the developed mathe-matical program with complementarity constrains, MPSOCC has many applications in practice. Motivated by the MPEC theory, several stationarity concepts, which include the Clarke-type, Mordukhovich-type, and strong stationarities, are presented in this chapter. It is further shown that a local minimizer of MPSOCC must be stationary in some sense under suitable conditions. This indicates that these stationarity concepts are reasonable in theory.