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arXiv · 2405.14637

On the role of semismoothness in nonsmooth numerical analysis: Theory

H. Gfrerer·J. V. Outrata
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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.

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H. Gfrerer, J. V. Outrata. 2026-03-11. On the role of semismoothness in nonsmooth numerical analysis: Theory. https://arxiv.org/abs/2405.14637

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