
Journal of Convex Analysis 15 (2008), No. 4, 831858 Copyright Heldermann Verlag 2008 A Complete Characterization of the Subdifferential Set of the Supremum of an Arbitrary Family of Convex Functions Abderrahim Hantoute Dept. of Statistics and Operations Research, University of Alicante, Apt. de Correos 99, 03071 Alicante, Spain hantoute@ua.es Marco Antonio López Dept. of Statistics and Operations Research, University of Alicante, Apt. de Correos 99, 03080 Alicante, Spain marco.antonio@ua.es Starting with some analysis of the support function of an arbitrary set, we obtain a formula for the subdifferential set of the supremum function of an arbitrary (possibly infinite) family of proper convex functions at each point of its effective domain, not necessarily at a continuity point. In this sense, our formula constitutes an extension of Theorem A of M. Volle ["Sousdifférentiel d'une enveloppe supérieure de fonctions convexes", Comptes Rendus Acad. Sci. Paris I 317 (1993) 845849], and also allows us to derive a generalization of a result A. Broensted ["On the subdifferential of the supremum of two convex functions", Math. Scand. 31 (1972) 225230]. Our approach is based on a linearization via the Fenchel conjugate. Keywords: Subdifferential set, support and supremum functions, convex analysis. MSC: 52A41, 90C25, 15A39 [ Fulltextpdf (212 KB)] for subscribers only. 