Multi-step nonlinear conjugate gradient methods for unconstrained minimization

Authors:
John A. Ford;Yasushi Narushima;Hiroshi Yabe
Affiliations:
Department of Computer Science, University of Essex, Colchester, UK CO4 3SQ;Department of Mathematical Information Science, Tokyo University of Science, Tokyo, Japan 162-8601;Department of Mathematical Information Science, Tokyo University of Science, Tokyo, Japan 162-8601
Venue:
Computational Optimization and Applications
Year:
2008

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Abstract

Conjugate gradient methods are appealing for large scale nonlinear optimization problems, because they avoid the storage of matrices. Recently, seeking fast convergence of these methods, Dai and Liao (Appl. Math. Optim. 43:87---101, 2001) proposed a conjugate gradient method based on the secant condition of quasi-Newton methods, and later Yabe and Takano (Comput. Optim. Appl. 28:203---225, 2004) proposed another conjugate gradient method based on the modified secant condition. In this paper, we make use of a multi-step secant condition given by Ford and Moghrabi (Optim. Methods Softw. 2:357---370, 1993; J. Comput. Appl. Math. 50:305---323, 1994) and propose two new conjugate gradient methods based on this condition. The methods are shown to be globally convergent under certain assumptions. Numerical results are reported.