Convergence analysis of sparse quasi-Newton updates withpositive definite matrix completion for two-dimensional functions

Yuhong Dai; Nobuo Yamashita

doi:10.3934/naco.2011.1.61

Article Contents

2011, Volume 1, Issue 1: 61-69. Doi: 10.3934/naco.2011.1.61

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Convergence analysis of sparse quasi-Newton updates with positive definite matrix completion for two-dimensional functions

Yuhong Dai^1, and
Nobuo Yamashita^2,

1.
State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, P.O. Box 2719, Beijing 100080
2.
Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501

Received: August 31, 2010

Revised: August 31, 2010

Published: January 2011

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Abstract

In this paper, we briefly review the extensions of quasi-Newton methods for large-scale optimization. Specially, based on the idea of maximum determinant positive definite matrix completion, Yamashita (2008) proposed a new sparse quasi-Newton update, called MCQN, for unconstrained optimization problems with sparse Hessian structures. In exchange of the relaxation of the secant equation, the MCQN update avoids solving difficult subproblems and overcomes the ill-conditioning of approximate Hessian matrices. A global convergence analysis is given in this paper for the MCQN update with Broyden's convex family assuming that the objective function is uniformly convex and its dimension is only two.
This paper is dedicated to Professor Masao Fukushima on occasion of his 60th birthday.

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Mathematics Subject Classification: Primary: 90C53, 90C06.

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