When some variational properties force convexity

Michel Volle (1) , Jean-Baptiste Hiriart-Urruty (2) , Constantin Zalinescu (3, 4)

1 LANLG - Laboratoire d'Analyse non linéaire et Géométrie
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 UAIC - Alexandru Ioan Cuza University of Iași = Universitatea Alexandru Ioan Cuza din Iași
4 Institute of Mathematics "Octav Mayer"

The notion of adequate (resp. strongly adequate) function has been recently introduced to characterize the essentially strictly convex (resp. essentially firmly subdifferentiable) functions among the weakly lower semicontinuous (resp. lower semicontinuous) ones. In this paper we provide various necessary and sufficient conditions in order that the lower semicontinuous hull of an extended real-valued function on a reflexive Banach space is essentially strictly convex. Some new results on nearest (farthest) points are derived from this approach.

Mots clés: convex duality; well posed optimization problem; essential strict convexity; essential smoothness; best approximation

Domaines: Mathématiques [math]

Michel Volle, Jean-Baptiste Hiriart-Urruty, Constantin Zalinescu. When some variational properties force convexity. ESAIM: Control, Optimisation and Calculus of Variations, 2013, 19 (3), pp.701-709. ⟨10.1051/cocv/2012029⟩. ⟨hal-00966249⟩