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Set-valued evenly convex functions: characterizations and c-conjugacy

M. D. Fajardo

TL;DR

This work extends evenly convex theory to set-valued maps $f:X \to \mathcal{P}(Z)$ under a cone-induced order $\leq_K$ in separated locally convex spaces. It shows that proper $K$-e-convex set-valued functions are the pointwise supremum of their set-valued $M_f$-affine minorants (along with $\mathcal{C}_f$-affine and $\mathcal{E}_f$ minorants) and develops a set-valued $c$-conjugation, yielding a Fenchel-Moreau–type biconjugation theorem $f^{cc'} = f_K$. The paper introduces and relates multiple minorant families—$M_f$, $\mathcal{H}_f$, $\mathcal{C}_f$, and $\mathcal{E}_f$—within a robust duality framework built on $K$-epigraphs and e-convex hulls. This provides a comprehensive duality theory for set-valued evenly convex optimization with potential applications to generalized quasiconvex programming and economic models in locally convex spaces.

Abstract

In this work we deal with set-valued functions with values in the power set of a separated locally convex space where a nontrivial pointed convex cone induces a partial order relation. A set-valued function is evenly convex if its epigraph is an evenly convex set, i.e., it is the intersection of an arbitrary family of open half-spaces. In this paper we characterize evenly convex set-valued functions as the pointwise supremum of its set-valued e-affine minorants. Moreover, a suitable conjugation pattern will be developed for these functions, as well as the counterpart of the biconjugation Fenchel-Moreau theorem.

Set-valued evenly convex functions: characterizations and c-conjugacy

TL;DR

This work extends evenly convex theory to set-valued maps under a cone-induced order in separated locally convex spaces. It shows that proper -e-convex set-valued functions are the pointwise supremum of their set-valued -affine minorants (along with -affine and minorants) and develops a set-valued -conjugation, yielding a Fenchel-Moreau–type biconjugation theorem . The paper introduces and relates multiple minorant families—, , , and —within a robust duality framework built on -epigraphs and e-convex hulls. This provides a comprehensive duality theory for set-valued evenly convex optimization with potential applications to generalized quasiconvex programming and economic models in locally convex spaces.

Abstract

In this work we deal with set-valued functions with values in the power set of a separated locally convex space where a nontrivial pointed convex cone induces a partial order relation. A set-valued function is evenly convex if its epigraph is an evenly convex set, i.e., it is the intersection of an arbitrary family of open half-spaces. In this paper we characterize evenly convex set-valued functions as the pointwise supremum of its set-valued e-affine minorants. Moreover, a suitable conjugation pattern will be developed for these functions, as well as the counterpart of the biconjugation Fenchel-Moreau theorem.
Paper Structure (2 sections, 1 theorem, 20 equations)

This paper contains 2 sections, 1 theorem, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries

Key Result

proposition thmcounterproposition

A set $C\subseteq X$ is e-convex if for every point $x_0\notin C$, there exists $x^*\in X^*$ such that $\left\langle x-x_0,x^*\right\rangle<0$, for all $x\in C$.

Theorems & Definitions (5)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • proposition thmcounterproposition
  • definition thmcounterdefinition