Newton methods for solving nonsmooth equations via a new subdifferential

A new subdifferential for a locally Lipschitzian function is proposed. Based on this subdifferential, Newton methods and inexact-Newton methods for solving the system of nonsmooth equations and for solving the system of equations of smooth compositions of nonsmooth functions, are developed. The Q-superlinear convergence of Newton methods and the Q-linear convergence of inexact-Newton methods are shown. The present Newton methods and inexact-Newton methods could be viewed as the extensions of previous ones with same convergent results.