The link between 1-norm approximation and effective positivstellensatze for the hypercube
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| Název: | The link between 1-norm approximation and effective positivstellensatze for the hypercube |
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| Autoři: | de Klerk, Etienne, Vera, J.C. |
| Zdroj: | de Klerk, E & Vera, J C 2024, 'The link between 1-norm approximation and effective positivstellensatze for the hypercube', Numerical Algebra Control and Optimization. https://doi.org/10.3934/naco.2024060 |
| Rok vydání: | 2024 |
| Sbírka: | Tilburg University: Research portal |
| Témata: | Polynomial kernel method, Positivstellensatz, Semidefinite programming |
| Popis: | The Schmudgen Positivstellensatz gives a certificate to verify positivity of a strictly positive polynomial f on a compact, basic, semi-algebraic set K subset of R-n. A Positivstellensatz of this type is called effective if one may bound the degrees of the polynomials appearing in the certificate in terms of properties of f. If K = [-1, 1](n) and 0 < f(min) := min(x is an element of K) f (x), then the degrees of the polynomials appearing in the certificate may be bounded by O(root f(max )- f(min/)f(min)), where f(max) := max(x is an element of K) f (x), as was recently shown by fmin Laurent and Slot [Optimization Letters 17:515-530, 2023]. The big-O notation suppresses dependence on n and the degree d of f. In this paper we show a similar result, but with a better dependence on n and d. In particular, our bounds depend on the 1-norm of the coefficients of f, that may readily be calculated. |
| Druh dokumentu: | article in journal/newspaper |
| Popis souboru: | application/pdf |
| Jazyk: | English |
| DOI: | 10.3934/naco.2024060 |
| Dostupnost: | https://research.tilburguniversity.edu/en/publications/80beadad-411c-4f16-a127-4e56b8d7ca96 https://doi.org/10.3934/naco.2024060 https://repository.tilburguniversity.edu/bitstreams/2cd0942a-857c-40d2-ba03-9bfe66a5aff2/download |
| Rights: | info:eu-repo/semantics/openAccess |
| Přístupové číslo: | edsbas.C3694984 |
| Databáze: | BASE |
Nájsť tento článok vo Web of Science | Abstrakt: | The Schmudgen Positivstellensatz gives a certificate to verify positivity of a strictly positive polynomial f on a compact, basic, semi-algebraic set K subset of R-n. A Positivstellensatz of this type is called effective if one may bound the degrees of the polynomials appearing in the certificate in terms of properties of f. If K = [-1, 1](n) and 0 < f(min) := min(x is an element of K) f (x), then the degrees of the polynomials appearing in the certificate may be bounded by O(root f(max )- f(min/)f(min)), where f(max) := max(x is an element of K) f (x), as was recently shown by fmin Laurent and Slot [Optimization Letters 17:515-530, 2023]. The big-O notation suppresses dependence on n and the degree d of f. In this paper we show a similar result, but with a better dependence on n and d. In particular, our bounds depend on the 1-norm of the coefficients of f, that may readily be calculated. |
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| DOI: | 10.3934/naco.2024060 |