On the Effective Putinar's Positivstellensatz and Moment Approximation

Abstract

We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar Positivstellensatz over a compact basic closed semialgebraic set S, with new polynomial bounds on the degree of the positivity certificates. These bounds involve a Łojasiewicz exponent associated to the description of S. We show that under Constraint Qualification Conditions, this Łojasiewicz exponent is equal to 1. We deduce new bounds on the convergence rate of the optima in Lasserre Sum-of-Squares hierarchy to the global optimum of a polynomial function on S and new bounds on the Hausdorff distance between the cone of truncated (probability) measures supported on S and the cone of truncated moment sequences, which are positive on the quadratic module of S.

Exact SOHS decompositions of trigonometric univariate polynomials with Gaussian coefficients

“…In the multivariate case, the situation is more delicate and providing degree bounds is already a challenge. Recent efforts have been pursued for polynomials positive over the unit sphere [11,31], or for more general closed sets of constraints defined by finitely many polynomials [3,26]. Theorem 2 in [11] provides us a degree bound that is linear in the number of variables while Theorem 1.7 in [3] and Corollary 1 in [26] give bounds that are polynomial in the input degrees and exponential in the number of variables.…”