Hybrid method for nonlinear least-square problems without calculating derivatives
type: Item
instance of: scholarly article
title: Hybrid method for nonlinear least-square problems without calculating derivatives
zbMATH Open document ID: 0677.90065
DOI: 10.1007/BF00939566
published in: Journal of Optimization Theory and Applications
publication date: 1990-01-01T00:00:00Z
review text: This paper presents a no-derivative modification of the hybrid Gauss- Newton-BFGS method for nonlinear least-square problems suggested initially by \textit{M. Al-Baali} and \textit{R. Fletcher} [J. Oper. Res. Soc. 36, 405-421 (1985; Zbl 0578.65064)] and modified later by \textit{R. Fletcher} and \textit{C. Xu} [IMA J. Numer. Anal. 7, 371-389 (1987; Zbl 0648.65051)]. The modification is made in such a way that, in a Gauss- Newton step, the Broyden's rank-one updating formula is used to obtain an approximate Jacobian and, in a BFGS step, the Jacobian is estimated using difference formulas. A set of numerical comparisons among the new hybrid method, the Gauss-Newton-Broyden method, and the finite-difference BFGS method is made and shows that the new hybrid method combines the better features of the Gauss-Newton-Broyden method and the finite-difference BFGS method. This paper also extends to the least-square problem the finite-termination property of the Broyden method, proved for a nonsingular system of equations by \textit{D. M. Gay} [SIAM J. Numer. Anal. 16, 623-630 (1979; Zbl 0453.65034)] and for the full-rank rectangular system of equations by \textit{R. R. Gerber} and \textit{F. T. Luk} [ibid. 18, 882-890 (1981; Zbl 0472.65036)].
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 65K05
zbMATH DE Number: 4110475
zbMATH Keywords: finite-termination property
zbMATH Keywords: no-derivative modification
zbMATH Keywords: hybrid Gauss- Newton-BFGS method
zbMATH Keywords: nonlinear least-square problems
zbMATH Keywords: rank-one updating formula
zbMATH Keywords: approximate Jacobian
author: Cheng-Xian Xu
describes a project that uses: NL2SOL
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00939566
OpenAlex ID: W2005068837
description: scientific article; zbMATH DE number 4110475