Newton Methods For Mathematical Programs With Conic Equilibrium Constraints

Mathematical programs with equilibrium constraints, or simply MPECs, are constrained optimization problems whose constraints include parameterized variational inequalities or parameterized generalized equations. Such problems are widely used in many fields such as economics and engineering. If the equilibrium constraints include parameterized generalized equations which are defined by closed convex cones, we call such MPECs mathematical programs with conic equilibrium constraints.This dissertation is devoted to the study of Newton methods for solving mathematical programs with conic equilibrium constraints, including Newton methods for solving mathematical programs governed by parameterized quasi-variational inequalities (QVIs), a smoothing Newton method for solving mathematical programs governed by second-order cone constrained generalized equations, and an inexact Newton method for solving mathematical programs with semidefinite cone complementarity constraints. The main results of this dissertation can be summarized as follows:1. In Chapter3, numerical methods for mathematical programs governed by QVIs are considered. The necessary optimality conditions for the optimization problem with QVI constraints are reformulated as a system of nonsmooth equations under the strict complementarity condition. The strongly BD-regularity for the nonsmooth system at its solution point is demonstrated under the linear independence constraint qualification and second order sufficient conditions. The smoothing Newton method is employed to solve this nonsmooth system. If the strict complementarity condition fails to hold, we can still reformulate the Mordukhovich (M-) stationary condition as a nonsmooth system. We introduce an inexact Newton method to solve this system and obtain the convergence under the same assumptions mentioned above.2. Chapter4focuses on a class of mathematical programs governed by second-order cone constrained parameterized generalized equations, we reformulate the necessary optimality conditions as a system of nonsmooth equations under the strict complementarity condition. For the purpose of convergence analysis, a set of second order sufficient conditions is proposed, which is proved to be sufficient for the second order growth. Then a smoothing Newton method is employed to solve this system whose strongly BD-regularity at a solution point is demonstrated under the linear independence constraint qualification and second order sufficient conditions.3. Chapter5is devoted to the study of mathematical programs with semidefinite cone complementarity constraints. First, we demonstrate an elegant formula for the tangent cone of semidefinite cone complementarity set, based on which the Bouligand (B-) stationary point is characterized explicitly. Then, the relationships among different stationary points under certain constraint qualifications are discussed. Finally, we propose a nonsmooth system reformulation of the stationary conditions under the strict complementarity condition and introduce an inexact Newton method to solve this system. For the purpose of convergence analysis, we propose a set of second order sufficient conditions for mathematical programs with semidefinite cone complementarity constraints, which together with the linear independence constraint qualification implies the local convergence.