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Journal of Convex Analysis 25 (2018), No. 4, 1253--1278
Copyright Heldermann Verlag 2018

Valadier-like Formulas for the Supremum Function I

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

Abderrahim Hantoute
DIM-CMM, Universidad de Chile, Beauchef 851, Santiago, Chile

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

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.

Keywords: Pointwise supremum function, convex functions, Fenchel subdifferential, Valadier-like formulas.

MSC: 26B05, 26J25, 49H05

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