Farkas-type theorems for positively homogeneous semi-infinite systems

Abstract

This article deals with systems of infinitely many inequalities involving functions that are positively homogeneous over a nonempty convex cone of the Euclidean space. Generalized convex conjugation theory is applied to derive a Farkas-type and a Gale-type theorem for this kind of systems. These results are particularized for linear and min-type inequality systems.

Keywords:

• Farkas-type theorems
• Positively homogeneous functions
• Generalized convex conjugation
• Semi-infinite inequality systems
• Linear systems
• Min-type systems
• Mathematics Subject Classifications 2000: 90C34
• 49N15

Acknowledgments

The research of the first author has been partially supported by the Spanish Ministry of Science and Technology, project BFM2002-04114-C02, and by the sabbatical program of Alicante University. The work of the second author has been supported by the Spanish Ministry of Science and Technology, project BEC2002-00642, and by the Departament d'Universitats, Recerca i Societat de la Informació, Direcció General de Recerca de la Generalitat de Catalunya, project 2001SGR-00162. He also thanks the support of the Barcelona Economics Program of CREA.

Notes

Dedicated to V.F. Demyanov on the occasion of his 65th birthday.

By X ⊂ 

to be a cone we mean that for every x ∈ X and λ > 0 one has λx ∈ X. Notice that, according to this definition, X need not contain the origin 0 _n .