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Article Dans Une Revue Optimization Letters Année : 2023

An effective version of Schmüdgen’s Positivstellensatz for the hypercube

Monique Laurent
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Lucas Slot

Résumé

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 ] .
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Dates et versions

hal-04089812 , version 1 (05-05-2023)

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Monique Laurent, Lucas Slot. An effective version of Schmüdgen’s Positivstellensatz for the hypercube. Optimization Letters, 2023, 17 (3), pp.515-530. ⟨10.1007/s11590-022-01922-5⟩. ⟨hal-04089812⟩

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