Nonmonotone Self-adaptive Levenberg-Marquardt Approach for Solving Systems of Nonlinear Equations

The well-known Levenberg-Marquardt method is used extensively to solve systems of nonlinear equations. An extension of the Levenberg-Marquardt method based on new nonmonotone technique is described. To decrease the total number of iterations, this method allows the sequence of objective function values to be nonmonotone, especially in the case where the objective function is ill-conditioned. Moreover, the parameter of Levenberg-Marquardt is produced according to the new nonmonotone strategy to use the advantages of the faster convergence of the Gauss-Newton method whenever iterates are near the optimizer, and the robustness of the steepest descent method in the case in which iterates are far away from the optimizer. The global and quadratic convergence of the proposed method is established. The results of numerical experiments are reported.