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Journal of Convex Analysis 17 (2010), No. 1, 321--348
Copyright Heldermann Verlag 2010

Convex Minimization Problems with Weak Constraint Qualifications

Christian Léonard
Modal-X -- Bât. G, Université Paris Ouest, 200 Av. de la République, 92001 Nanterre, France

One revisits the standard saddle-point method based on conjugate duality for solving convex minimization problems. Our aim is to reduce or remove unnecessary topological restrictions on the constraint set. Dual equalities and characterizations of the minimizers are obtained with weak or without constraint qualifications. The main idea is to work with intrinsic topologies which reflect some geometry of the objective function. The abstract results of this article are applied in other papers to the Monge-Kantorovich optimal transport problem and the minimization of entropy functionals.

Keywords: Convex optimization, saddle-point, conjugate duality.

MSC: 46N10, 49J45, 28A35

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