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New Quasi-Newton Equation and Related Methods for Unconstrained Optimization

New Quasi-Newton Equation and Related Methods for Unconstrained Optimization In unconstrained optimization, the usual quasi-Newton equation is B k+1 s k=y k, where y k is the difference of the gradients at the last two iterates. In this paper, we propose a new quasi-Newton equation, $$B_{k + 1} s_k = \tilde y_k $$ , in which $$\tilde y_k $$ is based on both the function values and gradients at the last two iterates. The new equation is superior to the old equation in the sense that $$\tilde y_k $$ better approximates ∇ 2 f(x k+1)s k than y k. Modified quasi-Newton methods based on the new quasi-Newton equation are locally and superlinearly convergent. Extensive numerical experiments have been conducted which show that the new quasi-Newton methods are encouraging. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Optimization Theory and Applications Springer Journals

New Quasi-Newton Equation and Related Methods for Unconstrained Optimization

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References (9)

Publisher
Springer Journals
Copyright
Copyright © 1999 by Plenum Publishing Corporation
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Optimization; Theory of Computation; Applications of Mathematics; Engineering, general; Operation Research/Decision Theory
ISSN
0022-3239
eISSN
1573-2878
DOI
10.1023/A:1021898630001
Publisher site
See Article on Publisher Site

Abstract

In unconstrained optimization, the usual quasi-Newton equation is B k+1 s k=y k, where y k is the difference of the gradients at the last two iterates. In this paper, we propose a new quasi-Newton equation, $$B_{k + 1} s_k = \tilde y_k $$ , in which $$\tilde y_k $$ is based on both the function values and gradients at the last two iterates. The new equation is superior to the old equation in the sense that $$\tilde y_k $$ better approximates ∇ 2 f(x k+1)s k than y k. Modified quasi-Newton methods based on the new quasi-Newton equation are locally and superlinearly convergent. Extensive numerical experiments have been conducted which show that the new quasi-Newton methods are encouraging.

Journal

Journal of Optimization Theory and ApplicationsSpringer Journals

Published: Sep 30, 2004

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