The Research On Nonlinear Conjugate Gradient Method And Global Convergence

Unconstrained optimization problem has wide application to many fields, especially in the command economy, engineering design, transport, production management, military defense and engineering technology etc. Therefore, the search for the optimization algorithm have important valve and meaning. The commonly methods, including steepest descent method, Newton’s method, quasi-Newton method, conjugate gradient method, etc, are used to solve unconstrained optimization problems.In the optimization algorithms, the steepest decent method has the advantages of small storage and simple structure when applied to unconstrained optimization but its convergence rate is too slow. The non-quasi-newton method applied to unconstrained optimization because of its faster convergence rate.And Conjugate gradient method is introduced by Hestence and Stiefel in 1952 in solving linear equations and is to be promoted non-linear optimization by Fletcher and Reeves in 1964.Conjugate gradient method is one of commonly used algorithm for solving unconstrained optimization problem. Because of the simplicity of their very low memory requirements and quadratic termination property etc. Based on the above advantages, therefore conjugate gradient method is widely used in large scale unconstrained optimization problems. With the development of the research, Conjugate gradient method have a lot of new research directions.Including hybrid conjugate gradient method, the memory of conjugate gradient method, spectral conjugate gradient method and the parameters of the conjugate gradient method, etc. In 2001, Bergin and Martinez have a spectral conjugate gradient method and combined with the spectral gradient method and conjugate gradient method. Spectral conjugate gradient method contains two directios regulatory parameters:spectrum parameters and conjugate parameters, is a kind of method that combines conjugate gradient method with spectral gradient method. In recent years. Research on conjugate gradient method has made considerable progress, but there are still shortcomings. Based on the previous scholars’ study. This paper mainly studies the concerning of the spectral conjugate gradient method. The main results obtained in this dissertation may be summarized as follows:In Chapter 1, the relevant concepts of non-linear conjugate gradient method is introduced. Then introduces iterative convergence and inexact line search And instruces the major results obtained in this paper.In Chapter 2, the background and the current status of research of the conjugate gradient method.Including the hybrid conjugate gradient method and the spectrum of conjugate gradient method, etc.Then several classic calculation formula is given.In Chapter 3, we present a new spectral conjugate gradient method by changing conjugate coefficient and spectrum coefficient. It is proved that the search direction is descent direction, and the global convergence property is established with the Wolfe line search.In Chapter 4, we present a new Hybrid spectral conjugate gradient method.It is proved that the search direction dk satisfies the sufficient descent considition. The objective function to meet convexity assumptions and the global convergence property is established with the Wolfe line search.