Some Studies On Theory And Algorithms For Mathematical Programs With Equilibrium Constraints

Mathematical program with equilibrium constraints, abbreviated as MPEC, is a con-strained optimization problem in which the essential constraints include some parametric variational inequality or parametric complementarity systems. One of the main sources of MPECs comes from the class of bilevel programming problems which have numerous ap-plications. Moreover, MPEC plays a very important role in many fields such as economic equilibria, game theory, engineering design, and transportation science. However, due to its constraints fail to satisfy standard constraint qualifications such as Mangasarian-Fromovitz constraint qualification at any feasible point, MPEC is very difficult to deal with from theoretical analysis and numerical methods.During the past two decades, MPEC has been extensively studied from theoretical analysis to numerical methods. However, there are still a lot problems worth studying. In this thesis, we will further study MPEC on theory and methods in depth. The main contribution are summarized as follows.(1) Although first-order optimality conditions and constraint qualifications for MPEC have been extensively studied, the weakest constraint qualifications for various sta-tionarities of MPEC have not been discussed in the literature. In Chapter2, we will study these weakest constraint qualifications in depth. More recently, a weak condition is given to show the isolatedness of stationary points, but it is not clear whether such a condition is a constraint qualification. In Chapter3, we show that such a condition is a constraint qualification for M-stationarity and it also implies the existence of a local error bound.(2) In Chapter3, we will study systematically the second-order optimality conditions for MPEC. We first study the second-order sufficient optimality condition for MPEC in terms of singular or nonsingular S-multipliers. Moreover, we introduce some weaker MPEC constraint qualifications and derive several second-order necessary optimality conditions for MPEC under these new MPEC constraint qualifications. Finally, we discuss the isolatedness of local minimizers and stationary points of MPEC under very weak conditions. (3) In Chapter4, we study stability for parametric mathematical programs with geomet-ric constraints, which is more general than parametric mathematical programs with equilibrium constraints. We show that, under some kind of constraint qualification and second-order sufficient condition or second-order growth condition, the locally optimal solution mapping and stationary point mapping are nonempty-valued and continuous with respect to the perturbation parameter and, under some suitable conditions, the stationary pair mapping is calm. Furthermore, we apply the new obtained results to several optimization problems in the literature. In particular, for MPEC, we show that the M-stationary pair mapping is calm with respect to the perturbation parameter if the M-multiplier second-order sufficient condition is satisfied, and the S-stationary pair mapping is calm if the S-multiplier second-order sufficient condition is satisfied and the bidegenerate index set is empty.(4) Chapter5aims at studying the sensitivity of parametric mathematical programs with equilibrium constraints. We establish the formulas for the directional deriva-tive of value functions under the perturbed MPEC-RCR regularity or the MPEC-NNAMCQ (they are both weaker than MPEC-LICQ). Moreover, we apply the new results to localized parametric mathematical programs with equilibrium con-straints, which reduces the required second-order sufficient conditions in literature. Finally, we investigate the subdifferential of value functions in terms of enhanced M-multipliers and enhanced C-multipliers.(5) Chapter6aims at developing effective numerical methods for solving mathematical programs with equilibrium constraints. Due to its constraints fail to satisfy stan-dard constraint qualifications, there exists various popular stationarity concept such as C-/M-/S-stationarity suggested in the literature. First we reformulate these sta-tionarity conditions as smooth equations with box constraints and then we present a modified Levenberg-Marquardt method for solving these constrained equations and then globalize this method. We show that, under some weak local error bound conditions, the method is globally convergent and, locally and superlinearly conver-gent. Finally, we discuss some sufficient conditions for local error bounds to hold and show that these conditions are not very stringent by a number of examples.(6) Chapter7aims at proposing an approach for obtaining normalized Nash stationary points in a class of equilibrium programs with equilibrium constraints (EPEC). We show that, under some kind of separable conditions of objective functions, the normalized C-/M-/S-Nash stationary points of EPEC can be calculated from solving an associated mathematical program with equilibrium constraints. In addition, we illustrate the application of the proposed approach by implementing it for solving two numerical examples in electricity markets.