On the Effective Putinar's Positivstellensatz and Moment Approximation

TL;DR

This paper provides a quantitative analysis of Putinar's Positivstellensatz, establishing new polynomial bounds on positivity certificates and convergence rates in polynomial optimization over semialgebraic sets.

Contribution

It introduces a new polynomial bound involving the Lojasiewicz exponent and applies it to derive convergence rates and Hausdorff distance bounds in moment optimization hierarchies.

Findings

New polynomial bound on positivity certificates involving Lojasiewicz exponent

First general polynomial bound on convergence rate of Lasserre's hierarchy

First Hausdorff distance bound between measure and pseudo-moment cones

Abstract

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 Lojasiewicz 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 Lojasiewicz 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.