Generalized Newton's Method based on Graphical Derivatives

TL;DR

This paper introduces a novel generalized Newton method utilizing graphical derivatives for solving nonsmooth nonlinear equations, achieving convergence without semismoothness assumptions, and expanding the applicability of Newton-type algorithms.

Contribution

The paper develops and justifies a new Newton algorithm based on graphical derivatives, a novel approach for nonsmooth equations, with proven convergence properties.

Findings

The algorithm exhibits local superlinear convergence.

The method converges globally under Kantorovich conditions.

It outperforms traditional semismooth Newton methods in examples.

Abstract

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness assumption, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and $B$-differentiable versions of Newton's method for nonsmooth Lipschitzian equations.