Valadier-like Formulas for the Supremum Function I

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.