Convergence Properties Of PRP Type Methods With Nonmonotone Line Searches

Conjugate gradient methods are a class of efficient algorithms for solving large-scale problems in optimization field because they possess some nice properties such as low memory storage and relatively fast convergence rate. The first conjugate gradient method is the linear conjugate gradient method proposed by Hestenes and Stiefel for solving the linear equations Ax= b in 1952. This method was extended to solving general nonlinear optimization problems by Fletcher and Reeves in 1964, which is called the FR method. So far, there are many nonlinear conjugate gradient methods. The PRP method has been regarded as one of the most efficient nonlinear conjugate gradient methods in numerical perspective. However, its convergence properties are not so satisfactory. The main reason lies in that it is not a descent method when the usual inexact line searches are used. Therefore, to improve its convergence properties, some modifications are often adopted or some new line searches are required.This paper studies convergence properties of two PRP type methods with some new nonmonotone line searches. Nonmonotone line search technique can improve effi-ciency of some algorithms since it is able to accept larger stepsizes, which makes the iterative sequence faster converge to some stationary point or optimal solution. The paper is organized as follows.In Chapter 1, we simply introduce the background of the problem and the related preliminary knowledge.In Chapter 2, we present a new nonmonotone line search and prove that the classical PRP method with the proposed line search converges globally for nonconvex optimization problems under suitable conditions.In Chapter 3, we establish the R-linear convergence rate of the PRP method with the line search proposed in Chapter2.In Chapter 4, we further study the modified PRP method introduced by Wei et al, and investigate its convergence properties. We prove that this modified PRP method has global convergence property under the strong Wolfe line search with σ=1/4 or some nonmonotone Armijo line search.In Chapter 5, we do some numerical experiments and report some numerical re-sults, which show that the PRP method with the proposed line search is efficient.