{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:26:12Z","timestamp":1787336772123,"version":"3.56.0"},"reference-count":41,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>It is known (see [J. M. Borwein and W. B. Moors, SIAM J. Optim., 8 (1998), pp. 309\u2013323], [R. Correa and A. Jofre, J. Optim. Theory Appl., 61 (1990), pp. 157\u2013174]) that the subdifferential of a semismooth or essentially smooth locally Lipschitz continuous function f over a Banach space determines this function up to an additive constant in the sense that any other function of the same type g whose subdifferential coincides with that of f at every point is equal to f plus a constant, i.e., $g=f+c$. Unfortunately, those classes of locally Lipschitz continuous functions do not include proper lower semicontinuous convex functions taking the value $+\\infty$ at some points. In this paper a new concept of essentially directionally smooth functions is introduced, and it is also shown, by a detailed analysis of enlarged inclusions of their subdifferentials, that these functions are subdifferentially determined up to an additive constant. It is also proved that the class of such functions contains proper lower semicontinuous convex functions and locally Lipschitz continuous functions which are arcwise essentially smooth. Moreover, it is established that the essentially directional smoothness property is preserved under addition. It is also shown that the class of essentially directionally smooth functions includes that of directionally stable functions (studied in [L. Thibault and D. Zagrodny, Canad. Math. Bull., 48 (2005), pp. 283\u2013301]) as well as several other classes of functions.<\/jats:p>","DOI":"10.1137\/090754571","type":"journal-article","created":{"date-parts":[[2010,5,26]],"date-time":"2010-05-26T18:52:18Z","timestamp":1274899938000},"page":"2300-2326","source":"Crossref","is-referenced-by-count":16,"title":["Subdifferential Determination of Essentially Directionally Smooth Functions in Banach Space"],"prefix":"10.1137","volume":"20","author":[{"given":"Lionel","family":"Thibault","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dariusz","family":"Zagrodny","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,5,26]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-02-03118-5"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2004.04.015"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-004-1174-z"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1006\/jfan.1997.3101"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623496297838"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623496305213"},{"key":"R7","unstructured":"M. Bounkhel,\n                      R\u00e9gularit\u00e9 tangentielle en analyse non lisse\n                      , Ph.D. thesis, Universit\u00e9 Montpellier 2, 1999."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1080\/02331930211992"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"F.H. Clarke,\n                      Optimization and Nonsmooth Analysis\n                      , Wiley Interscience, New York, 1983 (republished in 1990: Classics Appl. Math. 5, SIAM, Philadelphia).","DOI":"10.1137\/1.9781611971309"},{"key":"R10","first-page":"157","volume":"61","author":"Correa R.","year":"1990","journal-title":"J. Optim. Theory Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-3239","issn-type":"print"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1992-1126193-4"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1080\/01630569408816579"},{"key":"R13","first-page":"411","volume":"3","author":"Edmond J.F.","year":"2002","journal-title":"J. Nonlinear Convex Anal."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-247X(85)71120-1"},{"key":"R15","unstructured":"M. Geoffroy, F. Jules, and M. Lassonde,\n                      Integration of Subdifferentials of Lower Semicontinuous Functions\n                      , Preprint 00-02, Department of Mathematics, Universit\u00e9 des Antilles et de la Guyane, 2000."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1959.9.707"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1112\/S0025579300013541"},{"key":"R18","first-page":"5","volume":"54","author":"Ivanov M.","year":"2001","journal-title":"C. R. Acad. Bulgare Sci."},{"key":"R19","first-page":"117","volume":"294","author":"Janin R.","year":"1982","journal-title":"C. R. Acad. Sci. Paris"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1137\/0315061"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"B.S. Mordukhovich,\n                      Variational Analysis and Generalized Differentiation\n                      I, Grundlehren Math. Wiss. 330, Springer-Verlag, New York, 2006.","DOI":"10.1007\/3-540-31247-1"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"B.S. Mordukhovich,\n                      Variational Analysis and Generalized Differentiation\n                      II, Grundlehren Math. Wiss. 331, Springer-Verlag, New York, 2006.","DOI":"10.1007\/3-540-31246-3"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.24033\/bsmf.1625"},{"key":"R24","unstructured":"J.J. Moreau,\n                      Fonctionnelles Convexes\n                      , lecture notes, Coll\u00e8ge de France, 1967 (second edition published in 2003: Consiglio Nazionale delle Ricerche and Universit\u00e1 di Roma \u201cTor Vergata.\""},{"key":"R25","first-page":"155","volume":"2","author":"Ngai H.","year":"2000","journal-title":"J. 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