Nonmonotone globalization of inexact finite-difference Newton-iterative methods for nonlinear equations

In this paper we study nonmonotone globalization techniques, in connection with finite-difference inexact Newton-GMRES methods, for solving large-scale systems of nonlinear equations in the case that the Jacobian matrix is not available. We first define a globalization scheme, which combines nonmonotone watchdog rules and nonmonotone derivative-free line searches, and we prove its global convergence under the assumptions that the Jacobian is nonsingular and that the iterations of the GMRES subspace method can be completed. We also introduce a hybrid stabilization scheme employing line searches along positive bases, and we prove its global convergence under the assumption that the Jacobian is nonsingular. Numerical results on a set of large dimensional standard test problems show that the proposed techniques may give significant advantages in comparison with monotone implementations of Newton-type methods. 1This work was supported by MIUR, FIRB 2001 Research Program Large Scale Nonlinear Optimization, Roma, Italy.