  We consider the class of polynomial optimization problems $\inf \{f(x):x\in
K\}$ for which the quadratic module generated by the polynomials that define
$K$ and the polynomial $c-f$ (for some scalar $c$) is Archimedean. For such
problems, the optimal value can be approximated as closely as desired by
solving a hierarchy of semidefinite programs and the convergence is finite
generically. Moreover, the Archimedean condition (as well as a sufficient
coercivity condition) can also be checked numerically by solving a similar
hierarchy of semidefinite programs. In other words, under reasonable
assumptions the now standard hierarchy of SDP-relaxations extends to the
non-compact case via a suitable modification.
