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Journal of Convex Analysis 26 (2019), No. 1, 299--324
Copyright Heldermann Verlag 2019

Valadier-like Formulas for the Supremum Function. II: The Compactly Indexed Case

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

Continuing part I of this article of the same authors [Valadier-like formulas for the supremum function I, J. Convex Analysis 25(4) (2018) 1253--1278], we focus now on the compactly indexed case. We assume that the index set is compact and that the data functions are upper semicontinuous with respect to the index variable (actually, this assumption will only affect the set of ε-active indices at the reference point). As in the previous work, we do not require any continuity assumption with respect to the decision variable. The current compact setting gives rise to more explicit formulas, which only involve subdifferentials at the reference point of active data functions. Other formulas are derived under weak continuity assumptions. These formulas reduce to the 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 2] when the supremum function is continuous.

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

MSC: 26B05, 26J25, 49H05

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