Set-valued convex compositions

TL;DR

This paper derives a conjugation formula for the composition of set-valued functions in locally convex spaces, extending duality theory and providing a new tool for analyzing such functions.

Contribution

It introduces a conjugate formula for set-valued function compositions, advancing the duality theory in locally convex spaces.

Findings

Derived a conjugate formula for set-valued compositions

Provided dual representations under regularity conditions

Utilized Lagrange duality and minimax theory in proofs

Abstract

We study the composition of two set-valued functions defined on locally convex topological linear spaces. We assume that these functions map into certain complete lattices of sets that have been used to establish a conjugation theory for set-valued functions in the literature. Our main result is a formula for the conjugate of the composition in terms of the conjugates of the ingredient functions. As a special case, when the composition is proper and has further regularity, our formula yields a dual representation for the composition. The proof of the main result uses Lagrange duality and minimax theory in a nontrivial way.