Local convergence analysis of Newton's method for solving strongly regular generalized equations

TL;DR

This paper analyzes the local convergence of Newton's method for solving strongly regular generalized equations in Banach spaces, establishing superlinear and quadratic convergence under relaxed conditions using a majorant function.

Contribution

It introduces a unified analysis framework that relaxes Lipschitz conditions, providing optimal convergence radius and solution uniqueness for Newton's method in generalized equations.

Findings

Superlinear and quadratic convergence under strong regularity.

Relaxation of Lipschitz condition via a majorant function.

Unification of earlier Newton's method results.

Abstract

In this paper we study Newton's method for solving generalized equations in Banach spaces. We show that under strong regularity of the generalized equation, the method is locally convergent to a solution with superlinear/quadratic rate. The presented analysis is based on Banach Perturbation Lemma for generalized equation and the classical Lipschitz condition on the derivative is relaxed by using a general majorant function, which enables obtaining the optimal convergence radius, uniqueness of solution as well as unifies earlier results pertaining to Newton's method theory.