On the inexactness level of robust Levenberg–Marquardt methods

Abstract

Recently, the Levenberg–Marquardt (LM) method has been used for solving systems of nonlinear equations with nonisolated solutions. Under certain conditions it converges Q-quadratically to a solution. The same rate has been obtained for inexact versions of the LM method. In this article the LM method will be called robust, if the magnitude of the regularization parameter occurring in its sub-problems is as large as possible without decreasing the convergence rate. For robust LM methods the article shows that the level of inexactness in the sub-problems can be increased significantly. As an application, the local convergence of a projected robust LM method is analysed.

Keywords:

• nonlinear equation
• Levenberg–Marquardt method
• quadratic convergence
• nonisolated solution

Mathematics Subject Classifications 2000: :

• 65K05
• 90C30

Acknowledgements

The work of the second author was supported by a scholarship of the Gottlieb Daimler- und Karl Benz-Stiftung under Project No. 02-13/05.