Numerical Methods For Inverse Semi-definite Quadratic Programming Problems

We consider an inverse optimization problem raised from the semi-definite quadratic programming (SDQP) problem.In the inverse problem,the parameters in the objective function of a given SDQP problem are adjusted as little as possible so that a known feasible solution becomes the optimal one.We formulate this problem into a minimization problem with a positive semi-definite cone constraint,and its dual problem is a linearly positive semi-definite cone constrained semismoothly differentiable convex programming problem with fewer variables than the original one.Therefore this thesis is focused on solving the dual problem.The main results of this dissertation are summarized as follows:1.Chapter 1 first gives the introduction,background and current research of inverse optimization. Then we describe the inverse semi-definite quadratic programming problems studied in this thesis,and derive its dual problem ISDQD(A,B).2.Chapter 2 deals with the augmented Lagrangian method for the dual problem of the inverse semi-definite quadratic programming problems.In this chapter we first describe the background of the augmented Lagrangian method.Following a preliminary review of semismooth analysis and the positive semi-definite cone,we present the convergence analysis of augmented Lagrangian method for solving the dual problem.In the end,we report our numerical results.3.The contents of Chapter 3 are mainly about the smoothing Newton method for the Karush-Kuhn -Tucker system of problem ISDQD(A,B).First we introduce a smoothing function and study its properties.Then we present the convergence analysis of smoothing Newton method for the smooth linear system which is a smoothing approximation of the referred Karush-Kuhn-Tucker system.In the end of Chapter 3,our numerical experiment results show that this method is very effective.4.In Chapter 4,we discuss the inexact smoothing Newton method which is based on the smoothing Newton method.Following some preliminaries on matrix valued functions, we apply the inexact smoothing Newton method to solve problem ISDQD(A,B).Under the strict complementary and nondegeneracy assumptions,we give the convergence results of the inexact smoothing Newton method.Finally,in our numerical experiment we measure the numerical performance of four numerical methods for solving problem ISDQD(A,B).