Conjugate Duality of Set--Valued Functions

TL;DR

This paper introduces a conjugate duality framework for set-valued functions in pre-ordered locally convex spaces, using scalarizations and extending classical conjugation concepts to the set-valued context.

Contribution

It develops a Legendre-Fenchel conjugate for set-valued functions and links it with scalarizations and duality theories, advancing the mathematical foundation of set-valued analysis.

Findings

Scalarizations fully characterize set-valued functions.

A new Legendre-Fenchel conjugate for set-valued functions is defined.

Duality results connect conjugation with $(*,s)$--dualities.

Abstract

To a function with values in the power set of a pre--ordered, separated locally convex space a family of scalarizations is given which completely characterizes the original function. A concept of a Legendre--Fenchel conjugate for set-valued functions is introduced and identified with the conjugates of the scalarizations. The concept of conjugation is connected to the notion of $(*,s)$--dualities and duality results are provided.