A modification of the gradient method and function extremization
type: Item
instance of: scholarly article
title: A modification of the gradient method and function extremization
zbMATH Open document ID: 0687.65066
author: Tatsuo Noda
published in: Proceedings of the Japan Academy. Series A
publication date: 1989-01-01T00:00:00Z
review text: The aim of this paper is to present a modification of the gradient method generating the sequence \(\{x_ k\}\) satisfying \(x_{k+1}=x_ k-\alpha \nabla f(x_ k),\) where \(\alpha\) is a constant such that \[ (*)\quad 0<\alpha <2/\| H(\bar x)\|_ s, \] \(H(\bar x)\) being the Hessian matrix of f at \(\bar x,\) \(\bar x\) a stationary point of the objective function f and \(\| H(\bar x)\|_ s\) the spectral norm of the matrix \(H(\bar x)\). Since \(\bar x\) is unknown, in general it is very difficult to choose \(\alpha\) so that (*) holds. Therefore the author proposes an iteration method \[ x_{k+1}=x_ k-(\beta /\| H(x_ k)\|)\nabla f(x_ k), \] with a constant \(\beta\) satisfying \(0<\beta <2\). It is proved that the sequence of points \(\{x_ k\}\) generated in this way converges to a minimum point of the function f under the same conditions assumed for the original gradient method. A numerical example illustrating the efficiency of the presented modification is given too.
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 4126609
zbMATH Keywords: function extremization
zbMATH Keywords: convergence
zbMATH Keywords: gradient method
zbMATH Keywords: iteration method
zbMATH Keywords: numerical example
reviewed by: Nada Djuranović-Miličić
MaRDI profile type: Publication
full work available at URL: pjaa.65.39
OpenAlex ID: W2044430821
cites work: Q5644656
DOI: 10.3792/PJAA.65.39
description: scientific article; zbMATH DE number 4126609