On the convergence properties of the Levenberg–Marquardt method

Abstract

In this article, a new method is presented to update the parameter in the Levenberg–Marquardt Method (LMM) for solving nonlinear equation system, i.e., (namely, there exist positive constants c ₂ > 0, c ₃ > 0 such that ). The existing methods in [H. Dan, N. Yamashita and M. Fukushima (2001). Convergence Properties of the Inexact Levenberg-Marquardt Method Under Local Error Bound Conditions. Technical Report 2001-003. Department of Applied Mathematics and Physics, Kyoto University; N. Yamashita and M. Fukushima (2001). On the rate of convergence of the Levenberg-Marquardt method. Computing, 15, 239–249; J. Fan and Y. Yuan (2000). On the Convergence of a New Levenberg–Marquardt Method. Technical Report, State Key Laboratory of Scientific/Engineering Computing. Institute of Computational Mathematics and Scientific/Engineering Computing, CAS] are special cases of our method. We prove that the sequence generated by the method converges to the solution of the original equation system superlinearly and the exact order of convergence rate is if provides a local error bound for the system of nonlinear equations. It improves the existing results in [H. Dan, N. Yamashita and M. Fukushima (2001). Convergence Properties of the Inexact Levenberg-Marquardt Method Under Local Error Bound Conditions. Technical Report 2001-003. Department of Applied Mathematics and Physics, Kyoto University; N. Yamashita and M. Fukushima (2001). On the rate of convergence of the Levenberg-Marquardt method. Computing, 15, 239–249; J. Fan and Y. Yuan (2000). On the Convergence of a New Levenberg–Marquardt Method, Technical Report. State Key Laboratory of Scientific/Engineering Computing. Institute of Computational Mathematics and Scientific/Engineering Computing, CAS]. Furthermore, we generalize these results to nonlinear equation system with nonnegative constraints.

Keywords:

• Nonlinear equation system
• Levenberg–Marquardt Method
• Error bound
• Superlinear convergence
• Mathematics Subject Classifications 2000: 90C30
• 65K05
• 90C31
• 90C33

Acknowledgments

The author is grateful to the anonymous referees for their careful corrections and valuable suggestions and comments. This work is supported in part by National Science Foundation of China (Grant No. 10171055 and 70302003).