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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Subdifferential set
  • support and supremum functions
  • convex analysis

Mathematics Subject Classification

52A41, 90C25, 15A39

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Suggested citation

A. Hantoute, M. A. López. “A Complete Characterization of the Subdifferential Set of the Supremum of an Arbitrary Family of Convex Functions.” Journal of Convex Analysis 15 (2008), No. 4, 831–858. https://doi.org/10.68381/jca15054

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Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.