A Survey of Semidefinite Programming Approaches to the Generalized Problem of Moments and Their Error Analysis

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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“…The following result is shown in [10] on the convergence rate of the bound E (r) 1] n and consider the weight function w λ from (9).…”

Section: The Unit Cubementioning

confidence: 99%

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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“…When allowing decompositions in the preordering a stronger convergence rate in O(1/r ) was shown for special sets like the simplex (in [1]) and the hypercube (in [2]). (See also [7] for an overview). For the minimization of a homogeneous polynomial f over the unit sphere an improved convergence rate in O(1/r 2 ) for the bounds f (r ) was shown recently in [11] (improving the earlier rate in O(1/r ) from [9]).…”

Section: Previous Workmentioning

confidence: 99%

Near-optimal analysis of Lasserre’s univariate measure-based bounds for multivariate polynomial optimization

Slot

Laurent

2020

Math. Program.

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We consider a hierarchy of upper approximations for the minimization of a polynomial f over a compact set $$K \subseteq \mathbb {R}^n$$ K ⊆ R n proposed recently by Lasserre (arXiv:1907.097784, 2019). This hierarchy relies on using the push-forward measure of the Lebesgue measure on K by the polynomial f and involves univariate sums of squares of polynomials with growing degrees 2r. Hence it is weaker, but cheaper to compute, than an earlier hierarchy by Lasserre (SIAM Journal on Optimization 21(3), 864–885, 2011), which uses multivariate sums of squares. We show that this new hierarchy converges to the global minimum of f at a rate in $$O(\log ^2 r / r^2)$$ O ( log 2 r / r 2 ) whenever K satisfies a mild geometric condition, which holds, eg., for convex bodies and for compact semialgebraic sets with dense interior. As an application this rate of convergence also applies to the stronger hierarchy based on multivariate sums of squares, which improves and extends earlier convergence results to a wider class of compact sets. Furthermore, we show that our analysis is near-optimal by proving a lower bound on the convergence rate in $$\varOmega (1/r^2)$$ Ω ( 1 / r 2 ) for a class of polynomials on $$K=[-1,1]$$ K = [ - 1 , 1 ] , obtained by exploiting a connection to orthogonal polynomials.

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“…In fact, since this program has only one affine constraint, it even admits an eigenvalue reformulation [16], which will be mentioned in (12) in Section 2.2 below. Of course, in order to be able to compute the parameter (3) in practice, one needs to know explicitly (or via some computational procedure) the moments of the reference measure µ on K. These moments are known for simple sets like the simplex, the box, the sphere, the ball and some simple transforms of them (they can be found, e.g., in Table 1 in [10]).…”

Section: Introductionmentioning

confidence: 99%

“…This special case is already of independent interest, since it contains the problem of finding cubature schemes for numerical integration on the sphere, see e.g. [10] and the references therein. Our main result in Theorem 4 has the following implication for the GPM on the sphere, as a corollary of the following result in [13] (which applies to any compact K, see also [10] for a sketch of the proof in the setting described here).…”

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Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere

Klerk

Laurent

2020

Math. Program.

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We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp. 864 − 885], for the special case when the feasible set is the unit (hyper)sphere. The upper bound at level r ∈ N of the hierarchy is defined as the minimal expected value of the polynomial over all probability distributions on the sphere, when the probability density function is a sum-of-squares polynomial of degree at most 2r with respect to the surface measure.We show that the exact rate of convergence is Θ(1/r 2 ), and explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere a degree 3 polynomial over the unit sphere [20]:

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A Survey of Semidefinite Programming Approaches to the Generalized Problem of Moments and Their Error Analysis

Cited by 22 publications

References 58 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

Near-optimal analysis of Lasserre’s univariate measure-based bounds for multivariate polynomial optimization

Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere

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