Subsmooth sets: Functional characterizations and related concepts |
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Authors: | D Aussel A Daniilidis L Thibault |
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Institution: | Département de Mathématiques, Université de Perpignan, 66860 Perpignan Cedex, France ; Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra (Cerdanyola del Vallès), Spain ; Université Montpellier II, Département de Mathématiques, CC 051, Place Eugène Bataillon, 34095 Montpellier Cedex 05, France |
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Abstract: | Prox-regularity of a set (Poliquin-Rockafellar-Thibault, 2000), or its global version, proximal smoothness (Clarke-Stern-Wolenski, 1995) plays an important role in variational analysis, not only because it is associated with some fundamental properties as the local continuous differentiability of the function , or the local uniqueness of the projection mapping, but also because in the case where is the epigraph of a locally Lipschitz function, it is equivalent to the weak convexity (lower-C property) of the function. In this paper we provide an adapted geometrical concept, called subsmoothness, which permits an epigraphic characterization of the approximate convex functions (or lower-C property). Subsmooth sets turn out to be naturally situated between the classes of prox-regular and of nearly radial sets. This latter class has been recently introduced by Lewis in 2002. We hereby relate it to the Mifflin semismooth functions. |
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Keywords: | Variational analysis subsmooth sets submonotone operator approximately convex functions |
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