Geodesic
Valadier-like Formulas for the Supremum Function I
Journal of convex analysis, Tome 25 (2018) no. 4, pp. 1253-1278
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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.
Classification : 26B05, 26J25, 49H05
Mots-clés : Pointwise supremum function, convex functions, Fenchel subdifferential, Valadier-like formulas 
@article{JCA_2018_25_4_JCA_2018_25_4_a9,
     author = {R. Correa and A. Hantoute and M. A. L\'opez},
     title = {Valadier-like {Formulas} for the {Supremum} {Function} {I}},
     journal = {Journal of convex analysis},
     pages = {1253--1278},
     year = {2018},
     volume = {25},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JCA_2018_25_4_JCA_2018_25_4_a9/}
}
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R. Correa; A. Hantoute; M. A. López. Valadier-like Formulas for the Supremum Function I. Journal of convex analysis, Tome 25 (2018) no. 4, pp. 1253-1278. http://geodesic.mathdoc.fr/item/JCA_2018_25_4_JCA_2018_25_4_a9/