Two Typical Implementable Semismooth* Newton Methods for Generalized Equations are G-Semismooth Newton Methods

TL;DR

This paper reveals that two common implementations of Semismooth* Newton methods are actually specific cases of G-semismooth Newton methods, broadening theoretical understanding and aiding practical algorithm design for generalized equations.

Contribution

It establishes that two typical Semismooth* Newton methods are applications of G-semismooth Newton methods, enhancing theoretical insight and practical algorithm development.

Findings

Two typical implementations are exactly G-semismooth Newton methods.

This understanding broadens the theoretical framework of nonsmooth Newton methods.

Provides guidance for designing practical Newton-type algorithms.

Abstract

Semismooth* Newton methods have been proposed in recent years targeting multi-valued inclusion problems and have been successfully implemented to deal with several concrete generalized equations. In this paper, we show that two typical implementations of them that are available are exactly the applications of G-semismooth Newton methods for solving nonsmooth equations localized from these generalized equations. This new understanding expands the breadth of G-semismooth Newton methods in theory, results in a few interesting problems regarding the two categories of nonsmooth Newton methods, and more importantly, provides informative observations in facilitating the design and implementation of practical Newton-type algorithms for solving generalized equations.