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Journal Articles Mathematical Programming, Series A Year : 2022

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’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.
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Dates and versions

hal-03437328 , version 1 (19-11-2021)
hal-03437328 , version 2 (23-05-2022)
hal-03437328 , version 3 (24-08-2022)
hal-03437328 , version 4 (19-01-2023)

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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⟩
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