Strict convexity of sets in analytic terms

Dalla, Leoni; Hatziafratis, Telemachos

doi:10.1017/s1446788700014634

Cited by 3 publications

(4 citation statements)

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“…This assumption, however, is not sufficient. Instead we need the following stronger notion of 2-strict convexity introduced in [4]. Definition 4.…”

Section: Ball-like Convex Bodiesmentioning

confidence: 99%

“…It is shown in [4] that the set of 2-strictly convex bodies lies dense in the set of all convex bodies. For K with C 2boundary, it turns out that 2-strict convexity is equivalent to the existence of circumscribed tangent balls at all boundary points.…”

Section: Ball-like Convex Bodiesmentioning

confidence: 99%

“…For instance, for a = (0, −1), we have ∇Ψ(a) = (0, −4) and x T Ψ 2 (a)x = 12x 2 2 , which vanishes at x = (1, 0) ∈ T a K. In fact, one can verify that K does not have a circumscribed tangent ball at any of the points (0, ±1), (±1, 0). [4] that the set of 2-strictly convex bodies lies dense in the set of all convex bodies. For K with C 2boundary, it turns out that 2-strict convexity is equivalent to the existence of circumscribed tangent balls at all boundary points.…”

Section: Ball-like Convex Bodiesmentioning

confidence: 99%

“…We begin by recalling some of the basic properties of the Chebyshev polynomials. The Chebyshev polynomials T r ∈ R[t] r can be defined by the recurrence relation (4), and also by the following explicit expression:…”

Section: Needle Polynomialsmentioning

confidence: 99%

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Improved convergence analysis of Lasserre’s measure-based upper bounds for polynomial minimization on compact sets

Slot

Laurent

2020

Math. Program.

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We consider the problem of computing the minimum value f min,K of a polynomial f over a compact set K ⊆ R n , which can be reformulated as finding a probability measure ν on K minimizing K f dν. Lasserre showed that it suffices to consider such measures of the form ν = qµ, where q is a sum-of-squares polynomial and µ is a given Borel measure supported on K. By bounding the degree of q by 2r one gets a converging hierarchy of upper bounds f (r) for f min,K . When K is the hypercube [−1, 1] n , equipped with the Chebyshev measure, the parameters f (r) are known to converge to f min,K at a rate in O(1/r 2 ). We extend this error estimate to a wider class of convex bodies, while also allowing for a broader class of reference measures, including the Lebesgue measure. Our analysis applies to simplices, balls and convex bodies that locally look like a ball. In addition, we show an error estimate in O(log r/r) when K satisfies a minor geometrical condition, and in O(log 2 r/r 2 ) when K is a convex body, equipped with the Lebesgue measure. This improves upon the currently best known error estimates in O(1/ √ r) and O(1/r) for these two respective cases.Keywords polynomial optimization · sum-of-squares polynomial · Lasserre hierarchy · semidefinite programming · needle polynomial AMS subject classification 90C22; 90C26; 90C30 Improved convergence analysis of Lasserre's measure-based upper bounds for polynomial minimizationThe main disadvantage of the eigenvalue strategy is that it requires the moment matrix of f to have a closed form expression which is sufficiently structured so as to allow for an analysis of its eigenvalues. Closed form expressions for the entries of the matrix M r,f are known only for special sets K, such as the interval [−1, 1], the unit ball, the unit sphere, or the simplex, and only with respect to certain measures.However, as we will see in this paper, the convergence analysis from [10] in O(1/r 2 ) for the interval [−1, 1] equipped with the Chebyshev measure, can be transported to a large class of compact sets, such as the interval [−1, 1] with more general measures, the ball, the simplex, and 'ball-like' convex bodies.Analysis via the construction of feasible solutions. A second strategy to bound the convergence rate of the parameters E (r) (f ) is to construct explicit sum-of-squares density functions q r ∈ Σ r for which the integral K q r f dµ is close to f min,K . In contrast to the previous strategy, such constructions will only yield upper bounds on E (r) (f ).As noted earlier, the integral K f dν may be minimized by selecting the probability measure ν = δ a , the Dirac measure at a global minimizer a of f on K. When the reference measure µ is the Lebesgue measure, it thus intuitively seems sensible to consider sum-of-squares densities q r that approximate the Dirac delta in some way.This approach is followed in [12]. There, the authors consider truncated Taylor expansions of the Gaussian function e −t 2 /2σ , which they use to define the sum-of-squares polynomials

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“…This assumption, however, is not sufficient. Instead we need the following stronger notion of 2-strict convexity introduced in [4]. Definition 4.…”

Section: Ball-like Convex Bodiesmentioning

confidence: 99%

Section: Ball-like Convex Bodiesmentioning

confidence: 99%

Section: Ball-like Convex Bodiesmentioning

confidence: 99%

Section: Needle Polynomialsmentioning

confidence: 99%

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Improved convergence analysis of Lasserre’s measure-based upper bounds for polynomial minimization on compact sets

Slot

Laurent

2020

Math. Program.

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An Exact Radon Formula for Lambertian Tomography

Bellet

2022

J Math Imaging Vis

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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Characterizations of strictly convex sets by the uniqueness of support points

Hā

Jahn

2018

Optimization

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Strict convexity of sets in analytic terms

Cited by 3 publications

References 3 publications

Improved convergence analysis of Lasserre’s measure-based upper bounds for polynomial minimization on compact sets

Improved convergence analysis of Lasserre’s measure-based upper bounds for polynomial minimization on compact sets

An Exact Radon Formula for Lambertian Tomography

Characterizations of strictly convex sets by the uniqueness of support points

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