On the Levenberg-Marquardt methods for convex constrained nonlinear equations

Jinyan Fan

doi:10.3934/jimo.2013.9.227

Article Contents

2013, Volume 9, Issue 1: 227-241. Doi: 10.3934/jimo.2013.9.227

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On the Levenberg-Marquardt methods for convex constrained nonlinear equations

Jinyan Fan^1,

1.
Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240

Received: January 2012

Revised: July 2012

Published: January 2013

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Abstract

In this paper, both the constrained Levenberg-Marquardt method and the projected Levenberg-Marquardt method are presented for nonlinear equations $F(x)=0$ subject to $x\in X$, where $X$ is a nonempty closed convex set. The Levenberg-Marquardt parameter is taken as $\| F(x_k) \|_2^\delta$ with $\delta\in (0, 2]$. Under the local error bound condition which is weaker than nonsingularity, the methods are shown to have the same convergence rate, which includes not only the convergence results obtained in [12] for $\delta=2$ but also the results given in [7] for unconstrained nonlinear equations.

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Mathematics Subject Classification: 90C30, 65K05.

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