On Polynomial Optimization Over Non-compact Semi-algebraic Sets

We consider the class of polynomial optimization problems inf{f(x) : x ∈ 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.