  This paper considers polynomial optimization with unbounded sets. We give a
homogenization formulation and propose a hierarchy of Moment-SOS relaxations to
solve it. Under the assumptions that the feasible set is closed at infinity and
the ideal of homogenized equality constraining polynomials is real radical, we
show that this hierarchy of Moment-SOS relaxations has finite convergence, if
some optimality conditions (i.e., the linear independence constraint
qualification, strict complementarity and second order sufficiency conditions)
hold at every minimizer, including the one at infinity. Moreover, we prove
extended versions of Putinar-Vasilescu type Positivstellensatz for polynomials
that are nonnegative on unbounded sets. The classical Moment-SOS hierarchy with
denominators is also studied. In particular, we give a positive answer to a
conjecture of Mai, Lasserre and Magron in their recent work.
