Fenchel-Moreau identities on convex cones

TL;DR

This paper extends Fenchel-Moreau duality to functions on convex cones, providing conditions under which the biconjugation identity holds, especially for perfect cones, thus advancing convex analysis in ordered structures.

Contribution

It establishes Fenchel-Moreau identities for convex functions on cones, identifying conditions and classes of cones where these duality results are valid.

Findings

Fenchel-Moreau identity holds on certain convex cones.

Sufficient conditions identified for the identity to hold.

Perfect cones satisfy the conditions for the duality to apply.

Abstract

A pointed convex cone naturally induces a partial order, and further a notion of nondecreasingness for functions. We consider extended real-valued functions defined on the cone. Monotone conjugates for these functions can be defined in an analogous way to the standard convex conjugate. The only difference is that the supremum is taken over the cone instead of the entire space. We give sufficient conditions for the cone under which the corresponding Fenchel-Moreau biconjugation identity holds for proper, convex, lower semicontinuous, and nondecreasing functions defined on the cone. In addition, we show that these conditions are satisfied by a class of cones known as perfect cones.