A class of derivative-free methods for large-scale nonlinear monotone equations

In this paper, based on a line search technique proposed by Solodov and Svaiter (1998, Reformulation: Nonsmooth, Piecewise Smooth, Semismooth, and Smoothing Methods (M. Fukushima & L. Qi eds). Dordrecht: Kluwer, pp. 355-369), we propose a class of derivative-free methods for solving nonlinear monotone equations. These methods can be regarded as an extension of the spectral gradient method and some recently developed modified conjugate gradient methods for solving unconstrained optimization problems. Due to their lower storage requirement, these methods can be applied to solve large-scale nonlinear equations. We obtain global convergence of our methods without requiring differentiability, provided that the equation is Lipschitz continuous. Moreover, the whole sequence generated by the method converges to a solution of the equation even if the solution set is not a singleton. Preliminary numerical results show that the proposed methods are efficient.