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A new class of conjugate gradient methods with extended nonmonotone line search. (English) Zbl 1264.49019

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
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