Prox-regular sets and Legendre-Fenchel transform related to separation properties

Samir Adly (1) , Florent Nacry (2) , Lionel Thibault (3)

1 XLIM-MATHIS - Mathématiques & Sécurité de l'information
2 LAMPS - LAboratoire de Modélisation Pluridisciplinaire et Simulations
3 IMAG - Institut Montpelliérain Alexander Grothendieck

This paper is devoted to nonconvex/prox-regular separations of sets in Hilbert spaces. We introduce the Legendre-Fenchel r-conjugate of a prescribed function and r-quadratic support functionals and points of a given set, all associated to a positive constant r. By means of these concepts we obtain nonlinear functional separations for points and prox-regular sets. In addition to such functional separations, we also establish geometric separation results with balls for a prox-regular set and a strongly convex set.

Mots clés: Prox-regular set; strongly convex set; Legendre-Fenchel transform; semicon- vexity; quadratic support functional; geometric separation

Domaines: Mathématiques [math]; Optimisation et contrôle [math.OC]

Samir Adly, Florent Nacry, Lionel Thibault. Prox-regular sets and Legendre-Fenchel transform related to separation properties. 2020. ⟨hal-02567171⟩