GENERALIZATIONS OF THE HAHN-BANACH THEOREM REVISITED

Abstract

In this paper, based on the extended versions of the Farkas lemma for convex systems introduced recently in [9], we establish an extended version of a so called Hahn-Banach-Lagrange theorem introduced by Stephan Simons in [22]. This generalized version of the Hahn-Banach-Lagrange theorem holds in locally convex Hausdorff topological vector spaces under a Slater-type constraint qualification condition and with the relaxing of the lower semi-continuity of some functions involved and the closedness of the constrained sets. The version, in turn, yields extended versions of the Mazur-Orlicz theorem, the sandwich theorem, and the Hahn-Banach theorem concerning extended sublinear functions. It is then shown that all the generalized versions of the Farkas lemma for cone-convex/sublinear-convex systems in [9] and the new extended Hahn-Banach-Lagrange theorem just obtained are equivalent together. A class of composite problems involving sublinear-convex mappings is considered at the end of the paper. Here the main results of the paper are applied to get a strong duality result and optimality conditions for the class of problems. Moreover, a formula for the conjugate of the supremum of a family (possibly infinite, not lower semi-continuous) of convex functions is then derived from the duality result to show the generality and the significance of the class of problems in consideration.