A new version of the Liu–Storey conjugate gradient method q Chun-ming Tang a,b, * , Zeng-xin Wei b , Guo-yin Li b a College of Science, Shanghai University, 200444, Shanghai, China b College of Mathematics and Information Science, Guangxi University, 530004, Nanning, China Abstract In this paper, the global convergence of a new version of the Liu–Storey conjugate gradient method is discussed. This method combines the Liu–Storey conjugate gradient formula and a new inexact line search. We prove that the new method is globally convergent. Some preliminary numerical results show that the corresponding algorithm is efficient. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Conjugate gradient; Liu–Storey; Inexact line search; Nonconvex optimization; Global convergence 1. Introduction We consider the following unconstrained optimization problem: minff ðxÞjx 2 R n g; ð1:1Þ where f : R n ! R is a smooth and nonlinear function. During the last several decades, the research of the conjugate gradient methods was a very active topic due to the simplicity of their iteration and their excellent numerical performance in solving large scale optimization problems, see e.g. [15,7,11,3]. For the kth iterate x k , the conjugate gradient methods generate a new point x k+1 by x kþ1 ¼ x k þ k k d k ; ð1:2Þ where k k is a steplength determined by a certain line search, and d k is a search direction defined by d k ¼ g k ; if k ¼ 1; g k þ b k d k1 ; if k P 2;  ð1:3Þ 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.11.098 q Project supported by Chinese NSF Grants 10161002, Guangxi NSF Grants 0542043 and Guangxi University research foundation X061057. * Corresponding author. Address: College of Mathematics and Information Science, Guangxi University, 530004, Nanning, China. E-mail addresses: cmtang@gxu.edu.cn (C.-m. Tang), zxwei@gxu.edu.cn (Z.-x. Wei), gyli@math.cuhk.edu.hk (G.-y. Li). Applied Mathematics and Computation 189 (2007) 302–313 www.elsevier.com/locate/amc