Cubically Convergent Two-Step Gauss-Newton Method for Nonsmooth Equations with Application to AVE

Abstract

We propose a two-step Gauss-Newton method (TS-GNM) for solving nonsmooth equations. At each iteration, the TS-GNM solves both a Gauss-Newton equation and an approximate Gauss-Newton equation. A second-order derivative-free line search strategy is designed to ensure the global convergence of TS-GNM. Under the nonsingularity condition and the strong semismoothness of the underlying function, we prove that the TS-GNM converges quadratically. Furthermore, we demonstrate that the TS-GNM achieves a cubic convergence rate when the generalized Jacobian is locally Lipschitz continuous at the solutions. Finally, we pay particular attention to the absolute value equation and present some numerical results.