A Globally Convergence Spectral Conjugate Gradient Method for Solving Unconstrained Optimization Problems

Basim Hassan

doi:10.33899/csmj.2013.163543

Abstract

In this paper, a modified spectral conjugate gradient method for solving unconstrained optimization problems is studied, which has sufficient descent direction and global convergence with an inexact line searches. The Fletcher-Reeves restarting criterion was employed to the standard and new versions and gave dramatic savings in the computational time. The Numerical results show that the proposed method is effective by comparing it with the FR-method.

Highlights

Consider the unconstrained optimization problem min f (x) x R n .......... (1)where f : R n → R is continuously differentiable
Consider the conjugate gradient algorithm 2.1 where k and k are given by (8) and (3) respectively and k is obtained by the strong Wolfe line search (5) and (6)
Numerical Results we reported some numerical results obtained with the implementation of the new algorithm on a set of unconstrained optimization test problems

Summary

Introduction

Consider the unconstrained optimization problem min f (x) x R n where f : R n → R is continuously differentiable. For solution of (1), one of the algorithms in numerical performance is the Fletcher-Reeves (FR) conjugate gradient algorithm. Let g(x) denote the gradient of f at x, and x0 be an arbitrary initial approximate solution of (1). In a standard FR conjugate gradient algorithm, the search direction is determined by. A sequence of solutions will be generated by xk+1 = xk + k d k where k is the step length along dk+1 chosen by some kind of line search method and satisfies the strong Wolfe (SW ) conditions f We are going to develop a new conjugate gradient (CG) algorithm. The search direction generated by the method at each iteration satisfies the sufficient descent condition. We are going to establish the global convergence of the proposed algorithm with the Wolfe-type line search. The idea of CG methods had been studied by many researchers for example, see (Xiaoi et al, [6]); (Zhong et al, [7]) and (Zhang et al ,[8])

A New Conjugate Gradient Algorithm

Global Convergence

5.Conclusions and Discussions

Sum of Quartics function: