On Duality Theory in Vector Spaces without Topology via Algebraic Core

Abstract

In this paper, we study the Fenchel-Rockafellar duality and the Lagrange duality in the framework of vector spaces without topological structures via the algebraic core. We utilize the geometric approach, inspired from its successful application by B. S. Mordukhovich and his coauthors in variational and convex analysis (see [1,2,3,4,5,6]). First we establish conjugate calculus rules under qualifying conditions through the algebraic interior of the function’s domains. Then we develop sufficient conditions which guarantee the Fenchel-Rockafellar strong duality. Finally, after deriving some necessary and sufficient conditions for optimal solutions to convex minimization problems, under a Slater condition via the algebraic interior, we then obtain a sufficient condition for the Lagrange strong duality.