On the complexity of Putinar's Positivstellensatz

Jiawang Nie (1) , Markus Schweighofer (2)

1 USCD
2 IRMAR - Institut de Recherche Mathématique de Rennes

We prove an upper bound on the degree complexity of Putinar's Positivstellensatz. This bound is much worse than the one obtained previously for Schmüdgen's Positivstellensatz but it depends on the same parameters. As a consequence, we get information about the convergence rate of Lasserre's procedure for optimization of a polynomial subject to polynomial constraints.

Mots clés: Positivstellensatz; Complexity; Positive polynomial; Sum of squares; Quadratic module; Moment problem; Optimization of polynomials

Domaines: Géométrie algébrique [math.AG]; Algèbre commutative [math.AC]; Optimisation et contrôle [math.OC]

Jiawang Nie, Markus Schweighofer. On the complexity of Putinar's Positivstellensatz. Journal of Complexity, 2006, 23 (1), pp.135-150. ⟨10.1007/10.1016/j.jco.2006.07.002⟩. ⟨hal-00347086⟩