An adaptive damped Newton method for strongly monotone and Lipschitz continuous operator equations

TL;DR

This paper introduces an adaptive damped Newton method for solving strongly monotone and Lipschitz continuous operator equations, highlighting when undamped methods outperform damped ones and ensuring global convergence with adaptive step sizes.

Contribution

It presents a new adaptive step-size strategy for damped Newton methods that guarantees global convergence and clarifies the conditions favoring undamped updates.

Findings

Adaptive step-size strategy guarantees global convergence.

Undamped Newton method performs better near solutions.

The method is applicable to strongly monotone, Lipschitz continuous operators.

Abstract

We will consider the damped Newton method for strongly monotone and Lipschitz continuous operator equations in a variational setting. We will provide a very accessible justification why the undamped Newton method performs better than its damped counterparts in a vicinity of a solution. Moreover, in the given setting, an adaptive step-size strategy will be presented, which guarantees the global convergence and favours an undamped update if admissible.