Lagrange duality, stability and subdifferentials in vector optimization

TL;DR

This paper offers an alternative proof of strong duality in vector optimization problems using stability and subdifferential concepts, with dual variables being vectors, simplifying previous operator-based approaches.

Contribution

It introduces a new proof technique for strong duality in vector optimization, using vector dual variables and highlighting the simplicity over operator-based duality.

Findings

Duality with vectors is sufficient for strong duality results.

The space of dual variables matches the scalar case, simplifying the theory.

Operator duality is a straightforward extension of vector duality.

Abstract

Langrange duality theorems for vector and set optimization problems which are based on an consequent usage of infimum and supremum (in the sense greatest lower and least upper bounds with respect to a partial ordering) have been recently proven. In this note, we provide an alternative proof of strong duality for such problems via suitable stability and subdifferetial notions. In contrast to most of the related results in the literature, the space of dual variables is the same as in the scalar case, i.e., a dual variable is a vector rather than an operator. We point out that duality with operators is an easy consequence of duality with vectors as dual variables.