Fenchel-Rockafellar Theorem in Infinite Dimensions via Generalized Relative Interiors

TL;DR

This paper extends the Fenchel-Rockafellar duality theorem to infinite-dimensional locally convex topological vector spaces using generalized relative interiors, broadening its applicability beyond finite dimensions.

Contribution

It generalizes the classical Fenchel-Rockafellar theorem to infinite-dimensional spaces with new qualification conditions based on generalized relative interiors.

Findings

Proves Fenchel strong duality in LCTV spaces.

Introduces qualification conditions using generalized relative interiors.

Generalizes finite-dimensional duality results to infinite dimensions.

Abstract

In this paper we provide further studies of the Fenchel duality theory in the general frame work of locally convex topological vector (LCTV) spaces. We prove the validity of the Fenchel strong duality under some qualification conditions via generalized relative interiors imposed on the epigraphs and the domains of the functions involved. Our results directly generalize the classical Fenchel-Rockafellar theorem on strong duality from finite dimensions to LCTV spaces.