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arXiv · 2111.11258

On the Effective Putinar's Positivstellensatz and Moment Approximation

Lorenzo Baldi·Bernard Mourrain
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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.

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Lorenzo Baldi, Bernard Mourrain. 2021-11-22. On the Effective Putinar's Positivstellensatz and Moment Approximation. https://doi.org/10.1007/s10107-022-01877-6

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