Polynomial programming: LP-relaxations also converge

We consider the global minimization of a multivariate polynomial on a compact semialgebraic set. Using a result of Krivine [J. Analyse Math., 12 (1964), pp. 307-326; C. R. Acad. Sci. Paris, 258 (1964), pp. 3417-3418], it is shown that LP-relaxations based on products of the original constraints, in the spirit of the reformulation-linearization technique procedure of Sherali and Adams, converge to the global optimum. This extends to compact semialgebraic sets, previous results known for polytopes, and 0-1 programs. The asymptotic behavior of the LP-relaxations is also analyzed, and it is shown that in important cases, finite convergence cannot occur.