A note on a matrix version of the Farkas lemma

TL;DR

This paper investigates the minimal assumptions needed to extend a classical Farkas lemma to matrix polynomials, refining understanding of positivity representations in matrix polynomial optimization.

Contribution

It analyzes the assumptions in recent matrix polynomial Farkas lemmas, identifying which are essential for the results to hold.

Findings

Certain assumptions in the matrix Farkas lemma are unnecessary

The paper clarifies the conditions for matrix polynomial positivity representations

Results improve the theoretical foundation for matrix polynomial optimization

Abstract

A linear polyomial non-negative on the non-negativity domain of finitely many linear polynomials can be expressed as their non-negative linear combination. Recently, under several additional assumptions, Helton, Klep, and McCullough extended this result to matrix polynomials. The aim of this paper is to study which of these additional assumptions are really necessary.