Nonlinear Rescaling Methods For Nonconvex Second-Order Cone Programming Problems

Nonlinear Lagrangians are variants of the classical Lagrangian,in which the multipliers or constraint functions are involved in nonlinear ways.Nonlinear rescaling methods,based on a class of nonlinear Lagrangians,are important in constrained optimization.Meanwhile,second order cone programming problems have found wide applications in engineering, control and finance to robust optimization and combinatorial optimization.However, the study of numerical methods for nonconvex second order cone programming is not enough. Based on this observation,this dissertation focuses on the study of the rate of convergence of nonlinear rescaling methods for nonconvex second order cone programming.The main results obtained in this dissertation may be summarized as follows:1.Chapter 3 studies nordinear rescaling methods for solving nonconvex second order cone programming problems.Firstly,we discuss the differential properties of L(o|¨)wner operator and propose conditions on the real-valued function involved in the L(o|¨)wner operator to guarantee the convergence of the nonlinear rescaling methods.Secondly,we study the properties of the proposed class of nonlinear Lagrangians.Finally,under suitable conditions, we prove the rate of convergence of the proposed methods when the inner problems are exactly solved.The convergence theorem shows that the algorithm based on any of the proposed nonlinear Lagrangians in the class is locally convergent when the penalty parameter t is less than a threshold and the error bound of primal-dual solutions is proportional to t.Compared to the convergence analysis in nonlinear rescaling methods for nonlinear programming,we have to overcome the difficulty caused by the sigma term in the second order sufficient condition for nonconvex second order cone programming.2.Chapter 4 discusses the rate of convergence of the algorithms when the inner problems are approximately solved.Firstly,we give conditions on the real-valued function involved in the L(o|¨)wner operator to guarantee the convergence of the inexact algorithm.Secondly, we propose a stop criterion for an approximate solution of the inner problem.Finally,we establish the convergence theorem of the nonlinear rescaling method when this criterion is used.The convergence theorem shows that the algorithm based on any of the proposed nonlinear Lagrangians in the class is locally convergent when the penalty parameter t is less than a threshold and the error bound of primal-dual solutions is proportional to t.3.In Chapter 5 we verify that modified Frisch's function,modified Carroll's function,Log-Sigmoid function,MEC function and MFC function all satisfy the conditions proposed in Chapter 3 and Chapter 4.Base on these functions,we develop nonlinear rescaling methods and carry out the numerical experiments.The experiment results show that the methods are practical.