Journal of Convex Analysis 19 (2012), No. 2, 467--483
Copyright Heldermann Verlag 2012
Remarks on the Γ--regularization of Non-convex and Non-semi-continuous Functions on Topological Vector Spaces
Jean-Bernard Bru
Dep. de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain
jeanbernard_bru@ehu.es
Walter de Siqueira Pedra
Institut für Mathematik, Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany
pedra@mathematik.uni-mainz.de
We show that the minimization problem of any non-convex and non-lower semi-continuous function on a compact convex subset of a locally convex real topological vector space can be studied via an associated convex and lower semi-continuous function Γ(h). This observation uses the notion of Γ-regularization as a key ingredient. As an application we obtain, on any locally convex real space, a generalization of the Lanford III--Robinson theorem which has only been proven for separable real Banach spaces. The latter is a characterization of subdifferentials of convex continuous functions.
Keywords: Variational problems, non-linear analysis, non-convexity, Gamma-regularization, Lanford III -- Robinson theorem.
MSC: 58E30, 46N10, 52A07
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