Homotopy Methods For Nonlinear Programming, Nonlinear Second-Order Cone Programming And Nonlinear Semidefinite Programming

The nonlinear semidefinite programming problems have become an exciting branch of mathematical programming. They have important applications in many fields of science and en-gineering.Solving them is a major task of numerical optimization.Modern homotopy (interior point or non-interior point) method is a globally convergent and efficient method for optimiza-tion problems.In this dissertation,we improve combined homotopy interior point method for nonlinear programming,and consider homotopy methods to solve nonlinear second-order cone program-ming and nonlinear semidefinite programming problems.We consider the following questions:1.How to improve convergence conditions of combined homotopy interior point method for nonlinear programming;2.For nonlinear second-order cone programming and nonlinear semidefinite programming, whether there exists a good homotopy. If so,how to follow numerically the homotopy path to obtain a solution.In Chapter1,we give an introduction of homotopy methods,especially homotopy methods for solving nonlinear programming.Also we give an introduction of the theory as well as numer-ical methods for semidefinite programming,second-order programming, and their applications in the field of science and engineering.In Chapter2,a combined homotopy infeasible interior point method and constraint shift-ing homotopy method for solving nonlinear programming are given.In contrast to combined homotopy interior point method,proposed methods do not require starting from a feasible in-terior point,so they are more convenient to be implemented than combined homotopy interior point method.In addition, the normal cone conditions can be ensured in our method for some problems,for which the normal cone condition does not hold in combined homotopy interior point method.It is proven in theory that the constructed homotopies are good homotopies.And we make numerical comparison between our methods and LOQO6.01.By further numerical results,constraint shifting homotopy method and combined homotopy infeasible interior point method are competitive with LOQO6.01.Simple experiments show that our algorithm is feasi-ble and applicable. In Chapter3,a combined homotopy interior point method for solving nonlinear second-order cone programming problem (a class of nonlinear semidefinite programming) is given. The applicability of the proposed method is proven in theory:Existence and global convergence of the smooth homotopy path determined by constructed homotopy equation are given under nonemptiness and boundedness of the interior of the feasible set, a regularity assumption and the normal cone condition.Furthermore,better convergence results for convex second-order cone programming can be obtained.We prove some theoretical results and implement our method in Matlab programming language.And numerical results show its feasibility.In Chapter4,we present two homotopy methods for nonlinear semidefinite programming: constraint shifting homotopy method and aggregate homotopy method.We construct good ho-motopies directly by using parameterized matrix inequality constraints and KKT system of non-linear semidefinite programming.In the predictor-corrector algorithm following numerically the smooth homotopy path,AHO method is used to obtain a symmetric search direction.We give some theoretical results and implement the algorithm in Matlab programming language. The numerical experiments with nonlinear semidefinite programming formulations of several control design problems with the data contained in COMPleib are done.We make a compar-ison between our method and the existing methods to show their feasibility and applicability. We discuss the advantages of two homotopy methods in solving different classes of nonlinear semidefinite programming.