On the convergence of the modified Levenberg–Marquardt method with a nonmonotone second order Armijo type line search

Recently, Fan [J. Fan, The modified Levenberg–Marquardt method for nonlinear equations with cubic convergence, Math. Comput. 81 (2012) 447–466] proposed a modified Levenberg–Marquardt (MLM) method for nonlinear equations. Using a trust region technique, global and cubic convergence of the MLM method is proved by Fan (2012) [12] under the local error bound condition, which is weaker than nonsingularity. The purpose of the paper is to investigate the convergence properties of the MLM method with a line search technique. Since the search direction of the MLM method may be not a descent direction, standard line searches can not be used directly. In this paper, we propose a nonmonotone second order Armijo line search which guarantees the global convergence of the MLM method. Moreover, we prove that the unit step will be always accepted finally. Then cubic convergence of the MLM method is preserved under the local error bound condition. Some preliminary numerical results are also reported.