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Optim."],"published-print":{"date-parts":[[2026,9,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>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,\u00a0we consider not only semismooth derivatives of single-valued mappings but also their interplay with the semismoothness[Formula: see text] property for multifunctions.\u00a0In 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.\u00a0Further,\u00a0we show that semismooth derivatives coincide almost everywhere with generalized Jacobians and state some consequences concerning strict proto-differentiability for semismooth[Formula: see text] multifunctions.<\/jats:p>","DOI":"10.1137\/24m1663892","type":"journal-article","created":{"date-parts":[[2026,8,7]],"date-time":"2026-08-07T08:00:46Z","timestamp":1786089646000},"page":"1678-1705","source":"Crossref","is-referenced-by-count":0,"title":["On the Role of Semismoothness in Nonsmooth Numerical Analysis: Theory"],"prefix":"10.1137","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6860-102X","authenticated-orcid":true,"given":"Helmut","family":"Gfrerer","sequence":"first","affiliation":[{"name":"Johann Radon Institute for Computational and Applied Mathematics, A-4040 Linz, Austria, and Institute of Information Theory and Automation, Czech Academy of Sciences, 18208 Prague, Czech Republic."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ji\u0159\u00ed V.","family":"Outrata","sequence":"additional","affiliation":[{"name":"Institute of Information Theory and Automation, Czech Academy of Sciences, 18208 Prague, Czech Republic."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,8,7]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BF00938396"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-020-01501-5"},{"key":"ref3","volume-title":"Advances in Neural Information Processing Systems 34","author":"Bolte J.","year":"2021"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1137\/22M1541630"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2024.0617"},{"key":"ref6","volume-title":"Optimization and Nonsmooth Analysis","author":"Clarke F. 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