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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