On the convergence properties of the Levenberg-Marquardt method

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., mu(k) = Theta(dist(x(k), X*)(delta)) (0 < delta less than or equal to 2) (namely, there exist positive constants c(2) > 0, c(3) > 0 such that c(2)dist(x(k), X*)(delta) less than or equal to mu(k) less than or equal to c(3)dist(x(k), X*)(delta)). 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 min{1 + delta, 2} if \\F(x)\\ provides a local error bound for the system of nonlinear equations. It improves the existing results in [H. Dan, N. Yamashita and M. Fukushima (200 1). 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.