Farkas-type results for max-functions and applications

In the paper [9], Mangasarian introduced a new approach in order to give dual characterizations for different set containment problems. He succeeded to characterize the containment of a polyhedral set in another polyhedral set and in a reverse convex set defined by convex quadratic constraints and the containment of a general closed convex set in a reverse convex set defined by convex nonlinear constraints, respectively. By incorporating them as prior knowledge, these characterizations can be very useful in the determination of knowledge-based classifiers, the most famous example being here the so-called support vector machines classifiers. Motivated by the paper [9], Jeyakumar has established in [7] dual characterizations for the containment of a closed convex set, defined by infinitely many convex constraints, in an arbitrary polyhedral set, in a reverse convex set and in another convex set, respectively. The characterizations are given in terms of epigraphs of conjugate functions. Recently, Boţ and Wanka have presented in [3] some new Farkas-type results for inequality systems involving a finite as well as an infinite number of convex constraints. This approach bases on the theory of conjugate duality for convex optimization problems, namely by using the so-called Fenchel and Fenchel-Lagrange duality concepts (see also [1], [2], [10], [12]). Moreover the authors show how these new Farkas-type results generalize some of the results obtained by Jeyakumar in [7]. The aim of the present paper is to extend the results obtained in [3] by considering inequality systems involving finitely many convex constraints as well as convex max-functions. Then we particularize them in order to obtain set containment characterizations and, on the other hand, to rediscover two famous theorems of the alternative.