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Journal of Convex Analysis 13 (2006), No. 3, 687--694
Copyright Heldermann Verlag 2006

Global Maximum of a Convex Function: Necessary and Sufficient Conditions

Emil Ernst
Lab. de Modélisation en Mécanique et Thermodynamique, Université Paul Cezanne, Fac. des Sciences et Techniques, Casse 322, Av. Esc. Normandie-Niemen, 13397 Marseille

Michel Théra
LACO, Université de Limoges, 123 Avenue A. Thomas, 87060 Limoges, France

We prove that an extended-real-valued lower semi-continuous convex function Φ defined on a reflexive Banach space X achieves its supremum on every nonempty bounded and closed convex set of its effective domain Dom Φ, if and only if the restriction of Φ to Dom Φ is sequentially continuous with respect to the weak topology on the underlying space X.

Keywords: Global maximum of a convex function, optimality conditions.

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