A new geometric condition for Fenchel's duality in infinite dimensional spaces

Abstract

In 1951, Fenchel discovered a special duality, which relates the minimization of a sum of two convex functions with the maximization of the sum of concave functions, using conjugates. Fenchel's duality is central to the study of constrained optimization. It requires an existence of an interior point of a convex set which often has empty interior in optimization applications. The well known relaxations of this requirement in the literature are again weaker forms of the interior point condition. Avoiding an interior point condition in duality has so far been a difficult problem. However, a non-interior point type condition is essential for the application of Fenchel's duality to optimization. In this paper we solve this problem by presenting a simple geometric condition in terms of the sum of the epigraphs of conjugate functions. We also establish a necessary and sufficient condition for the e-subdifferential sum formula in terms of the sum of the epigraphs of conjugate functions. Our results offer further insight into Fenchel's duality.