{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:35:04Z","timestamp":1787340904101,"version":"3.56.0"},"reference-count":38,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Optim."],"published-print":{"date-parts":[[2013,1]]},"abstract":"<jats:p>Given a normed space $X$, we study the following problem: characterize the functions $f:X\\to\\mathbb{R}\\cup\\{+\\infty\\}$ such that for some $\\overline{x} \\in {\\rm dom}\\kern 0.12em f$ and some neighborhood $U$ of $\\overline{x}$, $\\partial f(x) \\subset Q\\ \\text{for all }x\\in U.$ The operator $\\partial$ denotes an abstract subdifferential, and $Q$ is a fixed subset of $X^*$ (the topological dual space of $X$). When $X$ is a Banach space and $f$ is lower semicontinuous, we provide a characterization via the support function of $Q$, based on the Zagrodny mean value theorem. If $Q=\\partial f(\\overline{x})$, the lower semicontinuous functions $f$ verifying the above subdifferential inclusion are positively homogeneous, up to some translation. Connections with the class of linearly well-conditioned functions are explored, and an application is given to the steepest descent equation. A parallel study is developed for an inequality on the directional derivative in place of the above subdifferential inclusion.<\/jats:p>","DOI":"10.1137\/120886212","type":"journal-article","created":{"date-parts":[[2013,3,26]],"date-time":"2013-03-26T11:41:07Z","timestamp":1364298067000},"page":"552-575","source":"Crossref","is-referenced-by-count":0,"title":["Inclusion of Subdifferentials, Linear Well-Conditioning, and Steepest Descent Equation"],"prefix":"10.1137","volume":"23","author":[{"given":"A.","family":"Cabot","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"L.","family":"Thibault","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2013,3,26]]},"reference":[{"key":"atypb1","first-page":"289","author":"Adly S.","year":"2006","journal-title":"New York"},{"key":"atypb2","unstructured":"H. Attouch, G. Buttazzo, and G. Michaille,<i>Variational Analysis in Sobolev and BV Spaces: Applications to PDE's and Optimization<\/i> MPS\/SIAM Ser. Optim. 6, SIAM, Philadelphia, 2006."},{"key":"atypb3","unstructured":"A. Auslender and M. Teboulle,<i>Asymptotic Cones and Functions in Optimization and Variational Inequalities<\/i> Springer Monogr. Math., Springer-Verlag, New York, 2003."},{"key":"atypb4","unstructured":"D. Az\u00e9,<i>El\u00e9ments d'analyse convexe et variationnelle<\/i> Ellipses, Paris, 1997."},{"key":"atypb5","first-page":"27","author":"Bishop E.","year":"1963","journal-title":"RI"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"J. F. Bonnans and A. Shapiro,<i>Perturbation Analysis of Optimization Problems<\/i> Springer Ser. Oper. Res., Springer-Verlag, New York, 2000.","DOI":"10.1007\/978-1-4612-1394-9"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"J. M. Borwein and A. Lewis,<i>Convex Analysis and Nonlinear Optimization<\/i> CMS Books Math.\/Ouvrages Math. SMC 3, Springer-Verlag, New York, 2000.","DOI":"10.1007\/978-1-4757-9859-3"},{"key":"atypb8","unstructured":"H. Br\u00e9zis,<i>Op\u00e9rateurs maximaux monotones dans les espaces de Hilbert et \u00e9quations d'\u00e9volution<\/i> Lecture Notes 5, North-Holland, Amsterdam, London, 1972."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/0022-1236(75)90027-0"},{"key":"atypb10","first-page":"439","volume":"31","author":"Burke J. V.","year":"2002","journal-title":"Control Cybernet."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/0331063"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-07-03990-6"},{"key":"atypb13","unstructured":"F. H. Clarke,<i>Optimization and Nonsmooth Analysis<\/i> John Wiley, New York, 1983."},{"key":"atypb14","unstructured":"F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski,<i>Nonsmooth Analysis and Control Theory<\/i> Grad. Texts in Math. 178, Springer-Verlag, New York, 1998."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1993-065-x"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022687412779"},{"key":"atypb17","first-page":"55","author":"Durier R.","year":"1988","journal-title":"Basel"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2006.12.038"},{"key":"atypb19","first-page":"765","volume":"17","year":"1980","journal-title":"Boll. Un. Mat. Ital. B (5)"},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"J.B. Hiriart-Urruty and C. Lemar\u00e9chal,<i>Convex Analysis and Minimization Algorithms,<\/i>I.<i>Fundamentals,<\/i>Springer-Verlag, Berlin, 1993.","DOI":"10.1007\/978-3-662-02796-7"},{"key":"atypb21","first-page":"153","author":"Lefebvre O.","year":"1988","journal-title":"Basel"},{"key":"atypb22","unstructured":"B. Lemaire,<i>Quelques r\u00e9sultats r\u00e9cents sur l'algorithme proximal<\/i> S\u00e9minaire d'Analyse Num\u00e9rique, Toulouse, 1989."},{"key":"atypb23","unstructured":"B. Lemaire,<i>About the Proximal Algorithm<\/i> Premier Colloque Franco-Sovi\u00e9tique de Programmation Math\u00e9matique, Luminy, 1990."},{"key":"atypb24","first-page":"39","author":"Lemaire B.","year":"1992","journal-title":"Berlin"},{"key":"atypb25","first-page":"71","volume":"12","author":"Lemaire B.","year":"1998","journal-title":"Pliska Stud. Math. Bulgar."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/09075367X"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-005-2667-0"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-005-4721-3"},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"B. S. Mordukhovich,<i>Variational Analysis and Generalized Differentiation,<\/i>I.<i>Basic Theory<\/i> Grundlehren Math. Wiss. 330, Springer-Verlag, Berlin, 2006.","DOI":"10.1007\/3-540-31247-1"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/BF02189795"},{"key":"atypb31","unstructured":"B. T. Polyak,<i>Sharp minima<\/i> presented at the International Workshop on Generalized Lagrangians and Their Applications, Laxenburg, Austria, 1979."},{"key":"atypb32","unstructured":"R. T. Rockafellar,<i>Convex Analysis<\/i> Princeton University Press, Princeton, NJ, 1970."},{"key":"atypb33","doi-asserted-by":"crossref","unstructured":"R. T. Rockafellar and R. Wets,<i>Variational Analysis<\/i> Springer, Berlin, 1998.","DOI":"10.1007\/978-3-642-02431-3"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1080\/02331939508844134"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1995.1003"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1985-058-1"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1016\/0362-546X(88)90088-0"},{"key":"atypb38","first-page":"272","author":"Z\u0103linescu C.","year":"2001","journal-title":"Russia"}],"container-title":["SIAM Journal on Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/120886212","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:55:26Z","timestamp":1787338526000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/120886212"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,1]]},"references-count":38,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2013,1]]}},"alternative-id":["10.1137\/120886212"],"URL":"https:\/\/doi.org\/10.1137\/120886212","relation":{},"ISSN":["1052-6234","1095-7189"],"issn-type":[{"value":"1052-6234","type":"print"},{"value":"1095-7189","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,1]]}}}