An effective version of Schmüdgen's Positivstellensatz for the hypercube

Abstract

Let S ⊆ R n be a compact semialgebraic set and let f be a polynomial nonnegative on S. Schmüdgen's Positivstellensatz then states that for any η > 0, the nonnegativity of f + η on S can be certified by expressing f + η as a conic combination of products of the polynomials that occur in the inequalities defining S, where the coefficients are (globally nonnegative) sum-of-squares polynomials. It does not, however, provide explicit bounds on the degree of the polynomials required for such an expression. We show that in the special case where S = [−1, 1] n is the hypercube, a Schmüdgen-type certificate of nonnegativity exists involving only polynomials of degree O(1/ √ η).This improves quadratically upon the previously best known estimate in O(1/η). Our proof relies on an application of the polynomial kernel method, making use in particular of the Jackson kernel on the interval [−1, 1].

Sum-of-squares hierarchies for polynomial optimization and the Christoffel-Darboux kernel

“…Alternatively, if η > 0, we have shown a bound in O(1/ √ η) on the degree of a Schmüdgen-type certificate of positivity for f −f min +η. Our result matches the recently obtained rates for the hypersphere [9] and the hypercube [18]. As a side result, we show similar convergence rates for the upper bounds (8) on these sets as well (w.r.t.…”

“…Comparison to the analysis on [−1, 1] n . For the analysis of the Schmüdgentype lower bounds on [−1, 1] n in [18], the authors make use of a multivariate Jackson kernel K jack 2r (see also [30]). This kernel is defined in terms of multivariate Chebyshev polynomials {C α : α ∈ N n } as: Putinar-type certificates.…”

“…For general compact semialgebraic sets X, the Schmüdgen-type bounds lb(f, T (X)) r converge to f min at a rate in O(1/r c ), where c > 0 again is a constant depending on X [23]. In the special case that X = [−1, 1] n is the hypercube, a convergence rate in O(1/r) was shown in [3], and recently improved to O(1/r 2 ) in [18]. When X = ∆ n , a rate in O(1/r) is known [11].…”

“…We give an overview of the technique we use to prove our main results Theorem 3 and Theorem 4. A similar technique was used to prove convergence results for other semialgebraic sets; on the hypersphere [9], on the binary hypercube {0, 1} n [26] and on the regular hypercube [−1, 1] n in [3] and [18]. Let f ∈ R[x] be a polynomial of degree d ∈ N. We wish to show that…”

On the Effective Putinar's Positivstellensatz and Moment Approximation

“…The ingredients for the proof are introduced in Section 2. The main differences with the one of [NS07] is the use of an effective Schmüdgen's Positivstellensatz on the unit box [LS21], and an effective approximation of regular functions on the unit interval, see Theorem 2.12. Moreover in Section 2.2 we prove that the main exponent of the bound, i.e.…”