Conservative and Semismooth Derivatives are Equivalent for Semialgebraic Maps

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