Global Convergence of a Spectral Conjugate Gradient Method for Unconstrained Optimization

Jinkui Liu,Youyi Jiang

doi:10.1155/2012/758287

Abstract

A new nonlinear spectral conjugate descent method for solving unconstrained optimization problems is proposed on the basis of the CD method and the spectral conjugate gradient method. For any line search, the new method satisfies the sufficient descent condition . Moreover, we prove that the new method is globally convergent under the strong Wolfe line search. The numerical results show that the new method is more effective for the given test problems from the CUTE test problem library (Bongartz et al., 1995) in contrast to the famous CD method, FR method, and PRP method.

Highlights

Unconstrained optimization problems have extensive applications, for example, in petroleum exploration, aerospace, transportation, and other domains
In order to establish the global convergence of our method, we need the following assumption on objective function, which have often been used in the literatures to analyze the global convergence of nonlinear conjugate gradient method and the spectral conjugate gradient method with inexact line searches
The strong Wolfe line search is a special case of the Wolfe line search, so the Lemma 2.6 holds under the strong Wolfe line search

Summary

Introduction

The strong Wolfe line search is a special case of the Wolfe line search, so the Lemma 2.6 holds under the strong Wolfe line search. We can use the same method to prove the Zoutendijk condition holding for the spectral conjugate gradient method. The following theorem establishes the global convergence of the new spectral conjugate gradient method with the strong Wolfe line search for the general functions. According to the given conditions, Lemma 2.6 all hold. We will obtain the conclusion 2.7 by contradiction. Suppose by contradiction that there exists a positive constant r > 0 such that gk ≥ r, 2.8 holds for ∀k ≥ 1. On the one hand, rewriting 1.11 as follows dk θkgk βkdk−1, 2.9 and squaring both side of it, we get dk 2 βk[2] dk−1 2 − 2θkgkT dk − θk[2] gk 2. · dk−1 2 − 2θkgkT dk − θk[2] gk 2

Global Convergence Analysis

Numerical Experiments