Positive polynomials on unbounded domains

TL;DR

This paper develops new certificates for polynomial non-negativity over unbounded domains, enabling convergent LMI hierarchies for polynomial optimization without the need for compactness assumptions.

Contribution

It characterizes non-negativity certificates on unbounded domains without quadratic modules and introduces copositive polynomials for broader, convergent optimization hierarchies.

Findings

Certificates valid on unbounded domains

Convergent LMI hierarchies for polynomial optimization

Use of copositive polynomials expands applicability

Abstract

Certificates of non-negativity such as Putinar's Positivstellensatz have been used to obtain powerful numerical techniques to solve polynomial optimization (PO) problems. Putinar's certificate uses sum-of-squares (sos) polynomials to certify the non-negativity of a given polynomial over a domain defined by polynomial inequalities. This certificate assumes the Archimedean property of the associated quadratic module, which in particular implies compactness of the domain. In this paper we characterize the existence of a certificate of non-negativity for polynomials over a possibly unbounded domain, without the use of the associated quadratic module. Next, we show that the certificate can be used to convergent linear matrix inequality (LMI) hierarchies for PO problems with unbounded feasible sets. Furthermore, by using copositive polynomials to certify non-negativity, instead of sos polynomials, the certificate allows the use of a very rich class of convergent LMI hierarchies to approximate the solution of general PO problems. Throughout the article we illustrate our results with various examples certifying the non-negativity of polynomials over possibly unbounded sets defined by polynomial equalities or inequalities.