Complexity Estimates for Representations of Schmüdgen Type

Mauch, Elizabeth

doi:10.1006/jcom.2001.0616

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2004

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“…Stengle obtained a similar bound even more concrete than ours for the special case where n ¼ 1 and S is a compact interval in R [19, Theorem 5] (see also [7]). Somehow related are also [13,Theorem 3.12] and [5,Theorem 1.2].…”

Section: Introductionmentioning

confidence: 61%

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On the complexity of Schmüdgen's Positivstellensatz

Schweighofer

2004

Journal of Complexity

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Schmu¨dgen's Positivstellensatz roughly states that a polynomial f positive on a compact basic closed semialgebraic subset S of R n can be written as a sum of polynomials which are non-negative on S for certain obvious reasons. However, in general, you have to allow the degree of the summands to exceed largely the degree of f : Phenomena of this type are one of the main problems in the recently popular approximation of non-convex polynomial optimization problems by semidefinite programs. Prestel (Springer Monographs in Mathematics, Springer, Berlin, 2001) proved that there exists a bound on the degree of the summands computable from the following three parameters: The exact description of S; the degree of f and a measure of how close f is to having a zero on S: Roughly speaking, we make explicit the dependence on the second and third parameter. In doing so, the third parameter enters the bound only polynomially. r

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Section: Introductionmentioning

confidence: 61%

“…The proof of the above lemma tells us how we could try to determine an appropriate c from given g 1 ; y; g m AR½ % X and e40: We have to compute r 1 ; y; r t fulfilling (7). This can be done using Gro¨bner bases [16,Section 4].…”

Section: Article In Pressmentioning

confidence: 97%

On the complexity of Schmüdgen's Positivstellensatz

Schweighofer

2004

Journal of Complexity

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Complexity Estimates for Representations of Schmüdgen Type

Cited by 1 publication

References 8 publications

On the complexity of Schmüdgen's Positivstellensatz

On the complexity of Schmüdgen's Positivstellensatz

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