On the subdifferential of the supremum of an arbitrary family of extended real-valued functions

• Autores: Marco A. López Cerdá, Michel Volle
• Localización: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas ( RACSAM ), ISSN-e 1578-7303, Vol. 105, Nº. 1, 2011, págs. 3-21
• Idioma: inglés
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• Resumen

  • In this paper, we derive a new formula for the subdifferential of the supremum of an arbitrary family of extended real-valued functions, in terms of the approximate subgradients of well chosen convex combinations of the data functions. The data functions are neither convex nor lower semicontinuous, but in this paper we assume that the supremum of the second conjugates of the data functions is proper and coincides with the second conjugate of the supremum function. Some applications of the main formula are provided. In particular, new formulas are given for the subdifferential of the closed convex hull of an extended real-valued function, as well as for the corresponding set of minimizers. We also get a generalization of the formula in Hiriart-Urruty et al. (J Funct Anal 118, 154-166 1993, Theorem 2.1) for the subdifferential of the sum. The paper finishes with the presentation of some open problems. © 2011 Springer-Verlag.