Classical linear vector optimization duality revisited

TL;DR

This paper revisits a classical vector dual problem in linear vector optimization, establishing duality results and clarifying its relation to other duals, while also characterizing efficient solutions under certain conditions.

Contribution

It reintroduces a neglected vector dual problem with a vector objective, proving duality theorems and connecting it to existing dual concepts in the literature.

Findings

Proves weak, strong, and converse duality for the vector dual problem.

Shows the equivalence of efficient and properly efficient solutions under certain conditions.

Clarifies the relationship between the classical dual and other dual frameworks.

Abstract

With this note we bring again into attention a vector dual problem neglected by the contributions who have recently announced the successful healing of the trouble encountered by the classical duals to the classical linear vector optimization problem. This vector dual problem has, different to the mentioned works which are of set-valued nature, a vector objective function. Weak, strong and converse duality for this "new-old" vector dual problem are proven and we also investigate its connections to other vector duals considered in the same framework in the literature. We also show that the efficient solutions of the classical linear vector optimization problem coincide with its properly efficient solutions (in any sense) when the image space is partially ordered by a nontrivial pointed closed convex cone, too.