| Mathematical programs with equilibrium constraints are constrained optimization problem-s whose constraints include parameterized variational inequalities or parameterized generalized equations.Such problems are widely used in many fields such as engineering design,economic equilibria and transportation science,data mining.However,due to the constraints,the feasible set violates most of the standard constraint qualifications such as Mangasarian-Fromovitz con-straint qualification.It has caused a lot of problems from the theoretical and numerical point of view.Therefore,one typically applies specialized algorithms in order to these problems.The regularization method is one prominent class of specialized algorithms.This dissertation is devoted to the study of the regularization method and numerical im-plementation for mathematical program with equilibrium constraints and mathematical program with vertical complementarity constraints.The basic idea of the regularization method is to re-place the original MPEC and MPVCC by a sequence of NLPs with a parameter such that the regularized problems approach the original one on parameter tending to zero.We improve the convergence properties of some regularization methods and design some feasible algorithms to solve them based on the inexact idea.The main results are summarized as follows:1.Chapter 3 concentrates on improving the convergence properties of the regularization schemes introduced by Kadrani et al.and Kanzow&Schwartz for MPEC by using the weaker MPEC relaxed constant positive-linear dependence.Furthermore,based on a NCP function,we present a proposed regularized problems for MPEC which has the hoped-for convergence results.We consider the semismooth Newton method and SQP method to solve the regularized problems.The numerical results indicate that the proposed regularization is an effective strategy for solving the MPEC.2.In chapter 4,we consider the inexact log-exponential regularization method for MPEC.We rewrite the complementary constraints as equality constraints,so that it has better conver-gence results under some approximate second-order conditions.The motivation of the approxi-mate stationary conditions is to define stopping criteria for many practical algorithms.A second-order primal-dual SQP method is presented to solve a sequence of log-exponential regularized problems.It is shown that an appropriate choice of termination conditions can guarantee to generate approximate second-order stationary points of regularized problems.Finally,the ex-perimental results verify the effectiveness of our approach.3.Chapter 5 focuses on the log-exponential regularization method for MPVCC.We consid-er the approximate KKT point of the regularized problem,since the idea of inexact KKT condi-tions can be used to define stopping criteria for many practical algorithms.We prove that,under the MPVCC-MFCQ assumption,the accumulation point of the inexact KKT points is Clarke sta-tionary point.Furthermore,we introduce a feasible strategy based on Newtonian penalty-barrier Lagrangian method that guarantees inexact KKT points.Numerical results show that the inexact regularization method is effective in solving MPVCC.4.Chapter 6 presents an approach to obtain the stationarity of MPVCC by solving a se-quence of Scholtes regularized problems.We consider the Scholtes regularization method with the sequence of inexact KKT points only,since it is realistic to compute inexact KKT points from a numerical point of view.For the purpose of convergence analysis,we propose the approxi-mate second-order conditions of Scholtes regularzed problem,which together with MPVCC linear independence constraint qualification obtains the M-stationarity.From these results,we apply an augmented Lagrangian method to obtain the inexact KKT points and give the conver-gence analy sis.In particular,the accumulation point of the generated iteration is a S-stationary point if some boundedness conditions hold.The numerical results show that it is an effective way to solve MPVCC. |
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