Theorems of the alternative and duality for inf-sup problems

The duality theory of convex mathematical programming is extended to inf-sup problems. To this end a new, general inf-sup theorem for two different convexlike, respectively concavelike payoff functions is established under an abstract closedness assumption, thus avoiding the usual compactness requirement. This closedness assumption is then made more concrete; in particular for the partially homogeneous programs under study, regularity conditions of Karlin and Slater type are discussed. Finally, related theorems of the alternative as well as a result of Hahn-Banach type are derived, where the usual bilinear form is replaced by a more general coupling function.