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

Abstract

Continuing with the work on the subdifferential of the pointwise supremum of convex functions, started in part I of this paper [R. Correa, A. Hantoute, M. A. Lopez, Valadier-like formulas for the supremum function I, J. Convex Analysis 25 (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 epsilon-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-differentiels d'une borne superieure et d'une somme continue de fonctions convexes, C. R. Acad. Sci. Paris Ser. A-B Math. 268 (1969) 39-42, Theorem 2], when the supremum function is continuous.