Local convergence of Newton's method for solving generalized equations with monotone operator

TL;DR

This paper analyzes the local quadratic convergence of Newton's method for solving generalized equations involving monotone operators in Hilbert spaces, unifying various convergence results through a majorant condition.

Contribution

It introduces a majorant condition framework to establish convergence, optimal radius, and rate for Newton's method applied to generalized equations with monotone operators.

Findings

Newton's method converges quadratically locally.

The convergence radius is explicitly characterized.

The approach unifies multiple existing convergence results.

Abstract

In this paper we study Newton's method for solving the generalized equation $F(x)+T(x)\ni 0$ in Hilbert spaces, where $F$ is a Fr\'echet differentiable function and $T$ is set-valued and maximal monotone. We show that this method is local quadratically convergent to a solution. Using the idea of majorant condition on the nonlinear function which is associated to the generalized equation, the convergence of the method, the optimal convergence radius and results on the convergence rate are established. The advantage of working with a majorant condition rests in the fact that it allow to unify several convergence results pertaining to Newton's method.