Geometric duality and parametric duality for multiple objective linear programs are equivalent

TL;DR

This paper demonstrates that parametric duality and geometric duality in multiple objective linear programming are equivalent, showing that each can be derived from the other through a simple geometric transformation.

Contribution

It proves the converse relationship, establishing that parametric duality can be derived from geometric duality, thus unifying the two duality theories.

Findings

Parametric duality can be derived from geometric duality.

A geometric transformation links the two duality theories.

Discussion of advantages of each duality approach.

Abstract

In 2011, Luc introduced parametric duality for multiple objective linear programs. He showed that geometric duality, introduced in 2008 by Heyde and L\"ohne, is a consequence of parametric duality. We show the converse statement: parametric duality can be derived from geometric duality. We point out that an easy geometric transformation embodies the relationship between both duality theories. The advantages of each theory are discussed.