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

Klerk, Etienne de; Laurent, Monique

doi:10.1137/100790835

Cited by 38 publications

(61 citation statements)

References 23 publications

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“…Here, c, c are constants (not explicitly known) depending only on K, and 2r is the selected degree bound. For the case of the hypercube, [5] shows (using Bernstein approximations) a convergence rate in O(1/r ) for the lower bounds based on Schmüdgen's Positivstellensatz.…”

Section: Introductionmentioning

confidence: 99%

Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization

2016

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We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre (SIAM J Optim 21(3):864-885, 2011), obtained by searching for an optimal probability density function h on K which is a sum of squares of polynomials, so that the expectationWe show that the rate of convergence is no worse than O(1/ √ r ), where 2r is the degree bound on the density function. This analysis applies to the case when f is Lipschitz continuous and K is a full-dimensional compact set satisfying some boundary condition (which is satisfied, e.g., for convex bodies). The r th upper bound in the hierarchy may be computed using semidefinite programming if f is a polynomial of degree d, and if all moments of order up to 2r +d of the Lebesgue measure on K are known, which holds, for example, if K is a simplex, hypercube, or a Euclidean ball.

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Section: Introductionmentioning

confidence: 99%

Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization

2016

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“…In [1], at the end of Sect. 4, the authors provide an explicit error bound for the quadratic case while assuming that the conjecture holds (the result is also reported in the third column of Table 1).…”

Section: Error Bounds For Putinar Type Representationsmentioning

confidence: 99%

“…In [1], DeKlerk and Laurent give explicit tight error bounds for p (r ) han,q and p (r ) sch,q : Theorem 2 (from Theorem 1.4 [1]) Let Q be described by the polynomials q 1 , . .…”

Section: Error Bounds For Schmüdgen Type Representationsmentioning

confidence: 99%

“…In [1,Sect. 3.2], the authors bound the constant C(d, p) using the fact that |a C (d, p), which yields the desired result.…”

Section: Lemma 3 For Every H Kmentioning

confidence: 99%

“…This work is a followup of [1], in which the authors use Bernstein multivariate approximation to derive some explicit error bounds for Schmüdgen/Handelman type representations. One of the concluding remarks of [1] is that some error bounds for Putinar type representation when optimizing quadratic polynomials over the hypercube can be obtained if the so-called "C n = 1 n(n+2) conjecture" is true.…”

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confidence: 99%

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Error bounds for polynomial optimization over the hypercube using putinar type representations

Magron

2014

Optim Lett

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Consider the optimization problem p min,Q := min x∈Q p(x), where p is a degree m multivariate polynomial and Q := [0, 1] n is the hypercube. We provide explicit degree and error bounds for the sums of squares approximations of p min,Q corresponding to the Positivstellensatz of Putinar. Our approach uses Bernstein multivariate approximation of polynomials, following the methodology of De Klerk and Laurent to provide error bounds for Schmüdgen type positivity certificates over the hypercube. We give new bounds for Putinar type representations by relating the quadratic module and the preordering associated with the polynomials g i := x i (1 − x i ), i = 1, . . . , n, describing the hypercube Q.

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Concluding Remarks

Li¹,

He²,

Zhang³

2012

SpringerBriefs in Optimization

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

Cited by 38 publications

References 23 publications

Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization

Convergence analysis for Lasserre’s measure-based hierarchy of upper bounds for polynomial optimization

Error bounds for polynomial optimization over the hypercube using putinar type representations

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