Two Typical Implementable Semismooth* Newton Methods for Generalized Equations are G-Semismooth Newton Methods
Liang Chen, Defeng Sun, Wangyongquan Zhang
TL;DR
The paper demonstrates that two practical semismooth$^*$ Newton algorithms for generalized equations are in fact instances of the G-semismooth Newton framework, applied to locally Lipschitz reformulations. By reformulating GE (2) with an auxiliary variable and GE (3) via a proximal residual, the authors establish Lipschitz localization, G-semismoothness, and regularity conditions that yield local superlinear convergence. This unifies previously proposed implementable methods under a single nonsmooth Newton theory, providing a solid foundation for developing robust Newton-type solvers for GEs and informing globalization strategies. The results extend the reach of G-semismooth Newton methods to generalized equations encountered in optimization and equilibrium modeling, with practical implications for large-scale nonsmooth problems.
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.