Researches On Some Mathematical Programs With Equilibrium Constraints

The mathematical program with equilibrium constraints (MPEC) is an optimization problem whose constraints include parameter variational inequality, complementarity problems and generalized equation. Such problems play an important role, for example, in mathematical economics, engineering design, chemical engineering, transportation networks, and so on. Meanwhile, the MPEC problem is closely related to variational inequality, Nash equilibrium and complementarity problems. However, the MPEC problem is generally difficult to deal with because their constraints fail to satisfy most of the standard qualifications. Especially, this kind of problems fail to satisfy the Mangasarian-Fromovitz constraint qualificication at any feasible point. During the past two decades, many researchers have been devoted to studying the theory analysis and the algorithm of the MPEC problem. However, there many problems which are needed to deal with about the MPEC problem. In our paper, based on the project function and smoothing function, we solve the different classes of MPEC problems and the Multiobjective Optimization Problems with Equilibrium Constraints with homotopy method. And we get the following solutions.1. We study the mathematical programs with whose constraints include box-constrained variational inequalities. Firstly, the variational inequality of the MPEC is transformed into an equality equation with projection function. To avoid the difficul-ty of the nonsmooth constraints, we smooth the projection function with Cabriel-Morefunction and establish the homotopy equation for the equality equation for the convenience of choose of the initial point. Without strongly monotone variational inequalities and adding additional multipliers, we get a sequence of approximate prob-lem of the MPEC problems.we prove that the solutions (stationary points) of the approximate problems converge to a solution (stationary point) of the original MPEC problem. The homotopy equation for the KKT system of the approximate problem and the existence and the convergence of the homotopy pathway are proven. Numerical experiments illustrate that the method is feasible and effective.2. We solve the mathematical program with nonlinear complementarity con-straints by homotopy method. Firstly, to avoid adding additional multipliers, we trans-form the original problem into a general nonlinear nonsmooth optimization problem and for the convenience the choose of the initial point, we construct the homotopy equation for the equation function of the approximate optimization problem. Finally, we construct homotopy equation for the KKT system of the approximate optimization problem. Meanwhile, we prove the existence and the convergence of the homotopy path. And it is shown that the KKT point of the approximate problem is the station-ary point of the initial problem. Numerical experiments illustrate that the method is feasible and effective.3. We use a new homotopy method for solving a class of multiobjective optimiza-tion problems with equilibrium constraints (MOPECs). By using sequentially bounded constraint qualifications, the MOPECs is reformulated as a general multiobjective op-timization with KKT system and the homotopy equation is constructed for the KKT system, then, we get a multiobjective optimization with inequality and equality. Fi-nally, we establish the homotopy equation for the equivalent problem of the MOPECs and prove the existence and convergence of the path that we track. Meanwhile, the numerical experiments indicate that the method is efficient.4. We solve the MPEC problem with the variational inequality constraints whose feasible set is a nonempty closed convex subset. The MPEC problem is reformulated as one-level optimization with projection function. Furthermore, we reform the MPEC problem as a one-level nonsmooth optimization problem. By the smoothing function we get a smooth approximate optimization of the MPEC. The homotopy equation is established for the KKT system of the approximate and the existence and the conver-gence of the homotopy path are proven.