On the Effective Putinar’s Positivstellensatz and Moment Approximation

Lorenzo Baldi (1) , Bernard Mourrain (1)

1 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes

We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar’s Positivstellensatz over a compact basic semialgebraic set S, with a new polynomial bound on the degree of the positivity certificates. This bound involves a Łojasiewicz exponent associated to the description of S. We show that if the gradients of the active constraints are linearly independent on S (Constraint Qualification condition), this Łojasiewicz exponent is equal to 1. We deduce the first general polynomial bound on the convergence rate of the optima in Lasserre’s Sum-of-Squares hierarchy to the global optimum of a polynomial function on S, and the first general bound on the Hausdorff distance between the cone of truncated (probability) measures supported on S and the cone of truncated pseudo- moment sequences, which are positive on the quadratic module of S.

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

Lorenzo Baldi, Bernard Mourrain. On the Effective Putinar’s Positivstellensatz and Moment Approximation. Mathematical Programming, Series A, 2022, ⟨10.1007/s10107-022-01877-6⟩. ⟨hal-03437328v4⟩