Nonlinear conjugate gradient methods with sufficient descent properties for unconstrained optimization

Wataru Nakamura; Yasushi Narushima; Hiroshi Yabe

doi:10.3934/jimo.2013.9.595

Article Contents

2013, Volume 9, Issue 3: 595-619. Doi: 10.3934/jimo.2013.9.595

This issue Previous Article On the robust control design for a class of nonlinearly affine control systems: The attractive ellipsoid approach Next Article Generalized weak sharp minima of variational inequality problems with functional constraints

Nonlinear conjugate gradient methods with sufficient descent properties for unconstrained optimization

1.
Central Japan Railway Company, JR Central Towers, 1-1-4, Meieki, Nakamura-ku, Nagoya, Aichi 450-6101
2.
Department of Management System Science, Yokohama National University, 79-4 Tokiwadai, Hodogaya-ku, Yokohama 240-8501
3.
Department of Mathematical Information Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601

Received: June 2012

Revised: March 2013

Published: July 2013

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Abstract

It is very important to generate a descent search direction independent of line searches in showing the global convergence of conjugate gradient methods. The method of Hager and Zhang (2005) satisfies the sufficient descent condition. In this paper, we treat two subjects. We first consider a unified formula of parameters which establishes the sufficient descent condition and follows the modification technique of Hager and Zhang. In order to show the global convergence of the conjugate gradient method with the unified formula of parameters, we define some property (say Property A). We prove the global convergence of the method with Property A. Next, we apply the unified formula to a scaled conjugate gradient method and show its global convergence property. Finally numerical results are given.

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Mathematics Subject Classification: Primary: 90C30, 90C06; Secondary: 65K05.

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