On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption
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.
scientific article; zbMATH DE number 2162513
scientific article