On the complexity of Schmüdgen's Positivstellensatz

Schweighofer, Markus

doi:10.1016/j.jco.2004.01.005

Cited by 76 publications

(82 citation statements)

References 17 publications

(46 reference statements)

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“…In this section we prove that given any > 0, there is a convex SDr set in sandwich between co(K) and co(K) + B and with an explicit SDr in terms of the g j 's that define K. For this purpose we use a result of Prestel (later refined by Schweighofer [18]) on a degree bound in Schmüdgen's Positivstellensatz (and similarly a result of Nie and Schweighofer [19] on a degree bound in Putinar's Positivstellensatz).…”

Section: Examples With Nonconvex Kmentioning

confidence: 99%

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Convex sets with semidefinite representation

Lasserre

2008

Math. Program.

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Abstract. We provide a sufficient condition on a class of compact basic semialgebraic sets K ⊂ R n for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials gj that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed > 0, there is a convex set K such that co(K) ⊆ K ⊆ co(K) + B (where B is the unit ball of R n ), and K has an explicit SDr in terms of the gj's. For convex and compact basic semi-algebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian L f associated with K and any linear f ∈ R[X] is a sum of squares. We also provide an approximate SDr specific to the convex case.

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Section: Examples With Nonconvex Kmentioning

confidence: 99%

“…Even more, f + ∈ P r (g) for some integer r ∈ N that does not depend on the precise value of f but only on its degree (here 1) and norm (here f = 1 and |f * | ≤ τ K ); see Schweighofer [18]. So let K be the SDr set defined in (2.25) with this r .…”

Section: In Both Cases (A) and (B) Bounds On R Are Availablementioning

confidence: 99%

Convex sets with semidefinite representation

Lasserre

2008

Math. Program.

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“…Some error bounds for the approximations obtained from the Positivstellensätze of Schmüdgen [26] and of Putinar [24] have been derived by Schweighofer [27] and by Nie and Schweighofer [21]. These bounds involve some constants that depend on the given data and are generally hard to compute.…”

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confidence: 99%

Error Bounds for Some Semidefinite Programming Approaches to Polynomial Minimization on the Hypercube

Klerk¹,

Laurent²

2010

SIAM J. Optim.

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Abstract. We consider the problem of minimizing a polynomial on the hypercube [0, 1] n and derive new error bounds for the hierarchy of semidefinite programming approximations to this problem corresponding to the Positivstellensatz of Schmüdgen [26]. The main tool we employ is Bernstein approximations of polynomials, which also gives constructive proofs and degree bounds for positivity certificates on the hypercube.

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“…For example, in the case of a compact set S, one has to take into account the degree, the size of the coefficients and the minimum of f on S, so be able to say something about the degree of the sums of squares (see [PD,Theorem 8.4.3] and [Sw1]). This is what makes it so difficult to find representations.…”

Section: Introductionmentioning

confidence: 99%

Stability of quadratic modules

Netzer

2009

manuscripta math.

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Abstract. A finitely generated quadratic module or preordering in the real polynomial ring is called stable, if it admits a certain degree bound on the sums of squares in the representation of polynomials. Stability, first defined explicitly in [PS], is a very useful property. It often implies that the quadratic module is closed; furthermore it helps settling the Moment Problem, solves the Membership Problem for quadratic modules and allows applications of methods from optimization to represent nonnegative polynomials.We provide sufficient conditions for finitely generated quadratic modules in real polynomial rings of several variables to be stable. These conditions can be checked easily. For a certain class of semi-algebraic sets, we obtain that the nonexistence of bounded polynomials implies stability of every corresponding quadratic module. As stability often implies the non-solvability of the Moment Problem, this complements the result from [Sch3], which uses bounded polynomials to check the solvability of the Moment Problem by dimensional induction. We also use stability to generalize a result on the Invariant Moment Problem from [CKS].

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On the complexity of Schmüdgen's Positivstellensatz

Cited by 76 publications

References 17 publications

Convex sets with semidefinite representation

Convex sets with semidefinite representation

Error Bounds for Some Semidefinite Programming Approaches to Polynomial Minimization on the Hypercube

Stability of quadratic modules

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