Abstract

We generalize and improve the original characterization given by M. Valadier [Sous-différentiels d'une borne supérieure et d'une somme continue de fonctions convexes, C. R. Acad. Sci. Paris, Sér. A-B Math. 268 (1969) 39–42; Theorem 1] of the subdifferential of the pointwise supremum of convex functions, involving the subdifferentials of the data functions at nearby points. We remove the continuity assumption made in that work and obtain a general formula for such a subdifferential. In particular, when the supremum is continuous at some point of its domain, but not necessarily at the reference point, we get a simpler version which gives rise to the Valadier formula. Our starting result is the characterization given by A. Hantoute, M. A. López and C. Zalinescu [Subdifferential calculus rules in convex analysis: a unifying approach via pointwise supremum functions, SIAM J. Optim. 19 (2008) 863–882; Theorem 4], which uses the ε-subdifferential at the reference point.

Author information

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

Rafael Correa

Universidad de O'Higgins, Rancagua, Chile
and: DIM-CMM, Universidad de Chile, Beauchef 851, Santiago, Chile

rcorrea@dim.uchile.cl

Marco Antonio López

Universidad de Alicante, 03080 Alicante, Spain
and: CIAO, Federation University, Ballarat, Australia

marco.antonio@ua.es

Keywords

  • Pointwise supremum function
  • convex functions
  • Fenchel subdifferential
  • Valadier-like formulas

Mathematics Subject Classification

26B05, 26J25, 49H05

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Suggested citation

R. Correa, A. Hantoute, M. A. López. “Valadier-like Formulas for the Supremum Function I.” Journal of Convex Analysis 25 (2018), No. 4, 1253–1278. https://doi.org/10.68381/jca25077

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press.