Prox-regular sets and Legendre-Fenchel transform related to separation properties. (English) Zbl 1493.49018
Summary: 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.
MSC:
| 49J52 | Nonsmooth analysis |
| 49J53 | Set-valued and variational analysis |
| 46N10 | Applications of functional analysis in optimization, convex analysis, mathematical programming, economics |
Keywords:
prox-regular set; strongly convex set; Legendre-Fenchel transform; semiconvexity; geometric separationReferences:
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