Scalar Representation and Conjugation of Set-Valued Functions

TL;DR

This paper introduces a scalarization approach for set-valued functions in pre-ordered locally convex spaces, defines a Legendre-Fenchel conjugate for these functions, and establishes duality results, advancing the theoretical framework of set-valued analysis.

Contribution

It presents a novel scalarization method and a Legendre-Fenchel conjugate for set-valued functions, linking them to scalar conjugates and deriving duality theorems.

Findings

Scalarizations fully characterize set-valued functions.

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

Weak and strong duality results are established.

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. Using this conjugate, weak and strong duality results are proven.