  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.
