Newton-type Methods For Solving Semi-infinite Programming Problems

Semi-infinite programming has wide applications in many fields such as engineering design,optimal control and economic equilibrium.In recent years it has become an active field of research in optimization.In this paper,we study the use of Newton type algorithms in semi-infinite programming.This paper is composed of three chapters.Chapter 1 is the introduction of this paper,which briefly introduces the development of semi-infinite programming and the main results in this paper.In chapter 2,we study a nonsmooth Levenberg-Marquardt method for solving semi-infinite programming problems.We first transform the semi-infinite programming problem into the KKT sys-tem by the techniques in[52,53].We extend the smoothing Levenberg-Marquardt algorithm for finite programming problems proposed by Dan,Fukushima,and Yamashita,in[54]and then propose a nonsmooth Levenberg-Marquardt algorithm for solving this system of semismooth equations.Comparing with[52,53],we remove the assumption that accmulation points exist and give a more general global convergence result.Furthermore,under the local error bound conditions,we prove that the nonsmooth Levenberg-Marquardt algorithm superlinearly(even quadratically) converges to a solution of semi-infinite programming problem.In the analysis of local convergence,it is worth noting that local error bound conditions are weaker than nonsingular conditions in[52,53](detail statement in[54]).In chapter 3,we study the use of Newton type algorithms in generalized scmi-infinitc programming.Wc first transform thc gcncralizcd scmi-infinitc programming problems into nonlinear equations under the constraint qualification of the extended Mangasarian-Fromowitz.We solve the generalized semi-infinite programming problem by the nonsmooth Levenberg-Marquardt algorithm proposed in chapter 2.Similiarly,we give global and local convergence analysis under the weaker condition.Finally,numerical results are given.