Liu, Hailin; Li, Xiaoyong A new class of conjugate gradient methods with extended nonmonotone line search. (English) Zbl 1264.49019 Appl. Math. Inf. Sci. 6, No. 1, Special Issue, 147-154 (2012). Summary: In this paper, we propose a new nonlinear conjugate gradient method for large-scale unconstrained optimization which possesses the following properties: (i) the sufficient descent condition \(-g_k^Td_k\geq\frac78\|g_k\|^2\) holds without any line searches; (ii) With exact line search, this method reduces to a nonlinear version of the Liu-Storey conjugate gradient scheme. (iii) Under some assumption, global convergence of this method is proved with a new nonmonotone line search. Preliminary numerical results show that this method is very efficient. MSC: 49K10 Optimality conditions for free problems in two or more independent variables 65K10 Numerical optimization and variational techniques 90C52 Methods of reduced gradient type Keywords:conjugate gradient; sufficient descent; hybrid method; unconstrained optimization PDFBibTeX XMLCite \textit{H. Liu} and \textit{X. Li}, Appl. Math. Inf. Sci. 6, No. 1, Special Issue, 147--154 (2012; Zbl 1264.49019)