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

Semismooth* Newton methods have been proposed in recent years targeting multivalued 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.

Funding: This work was supported by the National Key R&D Program of China [Grant 2021YFA1001300], the National Natural Science Foundation of China [Grant 12271150], the Natural Science Foundation of Hunan Province [Grant 2023JJ10001], the Science and Technology Innovation Program of Hunan Province [Grant 2022RC1190], the Hong Kong RGC Senior Research Fellow Scheme [Grant SRFS2223-5S02], and GRP Grants [15307822, 15307523].