An effective version of Schmüdgen's Positivstellensatz for the hypercube
Abstract
Let $S \subseteq \mathbb{R}^n$ be a compact semialgebraic set and let $f$ be a polynomial nonnegative on $S$. Schmüdgen's Positivstellensatz then states that for any $\eta > 0$, the nonnegativity of $f + \eta$ on $S$ can be certified by expressing $f + \eta$ 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 $O(1 / \sqrt{\eta})$. This improves quadratically upon the previously best known estimate in $O(1/\eta)$. Our proof relies on an application of the polynomial kernel method, making use in particular of the Jackson kernel on the interval $[-1, 1]$.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2021
- DOI:
- 10.48550/arXiv.2109.09528
- arXiv:
- arXiv:2109.09528
- Bibcode:
- 2021arXiv210909528L
- Keywords:
-
- Mathematics - Optimization and Control;
- 90C22;
- 90C23;
- 90C26
- E-Print:
- v2: implemented minor changes based on referee comments. 11 pages, 1 figure