On the Role of Semismoothness in Nonsmooth Numerical Analysis: Theory

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 their interplay with the semismoothness[Formula: see text] property for multifunctions. In particular, we are interested in semismooth derivatives of solution maps to parametric semismooth[Formula: see text] inclusions. Our results are expressed in terms of suitable generalized derivatives of the set-valued part, i.e., by limiting coderivatives or by subspace containing derivatives. Further, we show that semismooth derivatives coincide almost everywhere with generalized Jacobians and state some consequences concerning strict proto-differentiability for semismooth[Formula: see text] multifunctions.