An effective version of Schmüdgen’s Positivstellensatz for the hypercube

Abstract

Let $$S \subseteq \mathbb {R}^n$$ 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 $$\eta > 0$$ η > 0 , the nonnegativity of $$f + \eta$$ f + η on S can be certified by expressing $$f + \eta$$ 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 $$S = [-1, 1]^n$$ S = [ - 1 , 1 ] n is the hypercube, a Schmüdgen-type certificate of nonnegativity exists involving only polynomials of degree $$O(1 / \sqrt{\eta })$$ O ( 1 / η ) . This improves quadratically upon the previously best known estimate in $$O(1/\eta )$$ 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 $$[-1, 1]$$ [ - 1 , 1 ] .

Exponential convergence of sum-of-squares hierarchies for trigonometric polynomials

“…Therefore, existence of an SOS decomposition for f leads to the existence of the corresponding Schmüdgen representation for P on [−1, 1] d . Thus our results also provide convergence rates for this hierarchy, and we thus actually extend results from [10].…”

“…Proof We here follow the proof technique of [9,10] based on integral operators, by adapting it to trigonometric polynomials of degree 2r, which are easier to deal with than spherical harmonics or regular polynomials through the use of Fourier series. We consider the following integral operator on 1-periodic functions on [0, 1] d to R, defined as…”

“…• The proposed bound follows a series of earlier bounds with a similar behavior in O(1/s 2 ) for the convergence rate of Lasserre's SOS hierarchies, and uses the same proof technique based on integral operators [9,10,20]. The most closely related is the one of [10], which considers regular polynomials on [−1, 1] d with Schmüdgen's representation, but with a different choice for the function q in Eq. ( 3).…”

“…• We provide in Section 3 an O(1/s 2 ) convergence result for the level of the hierarchy corresponding to trigonometric polynomials of degree s, without any assumptions, that extends the work of [10] for polynomials on [−1, 1] d , with a similar proof technique (taken from [9]), but with simpler arguments and explicit constants.…”

“…However, finite convergence can only be shown when strict second-order local optimality conditions are satisfied, but without a bound on the level at which the finite convergence is achieved [8]. In terms of asymptotic convergence rates, they are quite slow, at best O(1/s 2 ) in the simplest situations for the relaxation with polynomials of degree s [9,10].…”