Continuity of Convex Set-valued Maps and a Fundamental Duality Formula for Set-valued Optimization

TL;DR

This paper explores various notions of continuity for set-valued maps in convex analysis, establishing a fundamental duality formula for set-valued optimization under the weakest continuity assumptions.

Contribution

It compares different continuity concepts for set-valued maps and derives a fundamental duality formula using the weakest continuity condition for regularity.

Findings

Comparison of prevalent continuity notions for set-valued maps.

Identification of the weakest continuity concept suitable for duality.

Derivation of a fundamental duality formula for set-valued optimization.

Abstract

Over the past years a theory of conjugate duality for set-valued functions that map into the set of upper closed subsets of a preordered topological vector space was developed. For scalar duality theory, continuity of convex functions plays an important role. For set-valued maps different notions of continuity exist. We will compare the most prevalent ones in the special case that the image space is the set of upper closed subsets of a preordered topological vector space and analyze which of the results can be conveyed from the extended real-valued case. Moreover, we present a fundamental duality formula for set-valued optimization, using the weakest of the continuity concepts under consideration for a regularity condition.