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An effective version of Schmüdgen's Positivstellensatz for the hypercube.

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Názov: An effective version of Schmüdgen's Positivstellensatz for the hypercube.
Autori: Laurent, Monique, Slot, Lucas
Zdroj: Optimization Letters; Apr2023, Vol. 17 Issue 3, p515-530, 16p
Abstrakt: 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 ] . [ABSTRACT FROM AUTHOR]
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Databáza: Complementary Index
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Abstrakt: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 ] . [ABSTRACT FROM AUTHOR]
ISSN:18624472
DOI:10.1007/s11590-022-01922-5