Journal of Convex Analysis 15 (2008), No. 4, 831--858
Copyright Heldermann Verlag 2008
A Complete Characterization of the Subdifferential Set of the Supremum of an Arbitrary Family of Convex Functions
Dept. of Statistics and Operations Research, University of Alicante, Apt. de Correos 99, 03071 Alicante, Spain
Marco Antonio López
Dept. of Statistics and Operations Research, University of Alicante, Apt. de Correos 99, 03080 Alicante, Spain
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 ["Sous-différentiel d'une enveloppe supérieure de fonctions convexes", Comptes Rendus Acad. Sci. Paris I 317 (1993) 845-849], 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) 225-230]. Our approach is based on a linearization via the Fenchel conjugate.
Keywords: Subdifferential set, support and supremum functions, convex analysis.
MSC: 52A41, 90C25, 15A39
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