On the role of semismoothness in nonsmooth numerical analysis: Theory

TL;DR

This paper explores the theoretical role of semismooth derivatives in nonsmooth numerical analysis, focusing on their properties, interplay with multifunctions, and implications for solution maps of parametric inclusions.

Contribution

It extends the theory of semismooth derivatives to multifunctions and solution maps, linking them with generalized derivatives and properties like strict proto-differentiability.

Findings

Semismooth derivatives coincide almost everywhere with generalized Jacobians.

The paper establishes conditions under which semismooth derivatives are applicable to solution maps.

Results provide a theoretical foundation for using semismooth derivatives in nonsmooth analysis.

Abstract

For the numerical solution of nonsmooth problems, sometimes it is not necessary that an exact subgradient/generalized Jacobian is at our disposal, but it suffices that a semismooth derivative, i.e., a mapping satisfying a certain semismoothness property, is available. In this paper we consider not only semismooth derivatives of single-valued mappings, but also its interplay with the semismoothness$^*$ property for multifunctions. In particular, we are interested in semismooth derivatives of solution maps to parametric semismooth$^*$ inclusions. Our results are expressed in terms of suitable generalized derivatives of the set-valued part, i.e., by limiting coderivatives or by SC (subspace containing) derivatives. Further we show that semismooth derivatives coincide a.e. with generalized Jacobians and state some consequences concerning strict proto-differentiability for semismooth$^*$ multifunctions.