On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption
type: Item
instance of: scholarly article
title: On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption
zbMATH Open document ID: 1076.65047
published in: Computing
publication date: 2005-04-29T00:00:00Z
review text: The problem of solving a system of nonlinear equations \(F(x)= 0\), where \(F:\mathbb{R}^n\to\mathbb{R}^m\) is continuously differentiable and \(F'\) satisfies a Lipschitz condition using the Levenberg-Marquardt method with the parameter updated from iteration to iteration is considered. Recently, \textit{N. Yamashita} and \textit{M. Fukushima} [Comput. Suppl. 15, 239--249 (2001; Zbl 1001.65047)] established an interesting quadratic convergence result or the Levenberg-Marquardt method without the nonsingularity assumption. This paper extends the result of Yamashita and Fukushima [loc. cit.] by using \(\mu_k= \| F(x_k)\|^\delta\), where \(\delta\in [1,2]\), instead of \(\mu_k= \| F(x_k)\|^2\) as the Levenberg-Marquardt parameter. If \(\| F(x)\|\) provides a local error bound for the system of nonlinear equations \(F(x)= 0\), it is shown that the sequence \(\{x_k\}\) generated by the new method converges to a solution quadratically, which is stronger than \(\text{dist}(x_k, X^*)\to 0\) given by Yamashita and Fukushima. Numerical results show that the method performs well for singular problems.
reviewed by: Jiří Vaníček
Mathematics Subject Classification ID: 65H10
zbMATH DE Number: 2162513
zbMATH Keywords: system of nonlinear equations
zbMATH Keywords: Levenberg-Marquardt method
zbMATH Keywords: quadratic convergence
zbMATH Keywords: error bound
zbMATH Keywords: numerical results
author: Jinyan Fan
author: Yaxiang Yuan
describes a project that uses: minpack
describes a project that uses: KELLEY
MaRDI profile type: Publication
full work available at URL: s00607-004-0083-1
OpenAlex ID: W2057418187
cites work: A method for the solution of certain non-linear problems in least squares
cites work: An Algorithm for Least-Squares Estimation of Nonlinear Parameters
cites work: Testing Unconstrained Optimization Software
cites work: Q3840111
cites work: An Iterative Method for Finding Stationary Values of a Function of Several Variables
cites work: Tensor Methods for Nonlinear Equations
cites work: Q3998482
cites work: Q2765625
cites work: Q4538036
cites work: A review of trust region algorithms for optimization
DOI: 10.1007/S00607-004-0083-1
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description: scientific article; zbMATH DE number 2162513