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

Abstract

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.