A Complete Characterization of the Subdifferential Set of the Supremum of an Arbitrary Family of Convex Functions

Abstract

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. Brøndsted ["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.