Conservative and semismooth derivatives are equivalent for semialgebraic maps

TL;DR

This paper proves that for semi-algebraic maps, conservative and semismooth derivatives are equivalent, simplifying the understanding of generalized derivatives used in nonsmooth optimization algorithms.

Contribution

It establishes the equivalence of conservative and semismooth derivatives in the semi-algebraic setting, unifying two important concepts in nonsmooth analysis.

Findings

Conservative and semismooth derivatives coincide for semi-algebraic maps.

A new proof shows semi-algebraic maps are semismooth relative to the Clarke Jacobian.

The results simplify the analysis of generalized derivatives in nonsmooth optimization.

Abstract

Subgradient and Newton algorithms for nonsmooth optimization require generalized derivatives to satisfy subtle approximation properties: conservativity for the former and semismoothness for the latter. Though these two properties originate in entirely different contexts, we show that in the semi-algebraic setting they are equivalent. Both properties for a generalized derivative simply require it to coincide with the standard directional derivative on the tangent spaces of some partition of the domain into smooth manifolds. An appealing byproduct is a new short proof that semi-algebraic maps are semismooth relative to the Clarke Jacobian.