Conservative and semismooth derivatives are equivalent for semialgebraic maps

Understand

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.