Set-valued approximations and Newton's methods

We introduce a point-based set-valued approximation for a mapping from R-n to R-m. Under the assumption of semi-smoothness of the mapping, we prove that the approximation can be obtained through the Clarke generalized Jacobian, Ioffe-Ralph generalized Jacobian, B-subdifferential and their approximations. As an application, we propose a generalized Newton's method based on the point-based set-valued approximation for solving nonsmooth equations. We show that the proposed method converges locally superlinearly without the assumption of semi-smoothness. Finally we include some well-known generalized Newton's methods in our method and consolidate the convergence results of these methods.