Worst-Case Examples for Lasserre’s Measure–Based Hierarchy for Polynomial Optimization on the Hypercube

2020

Math. Program.

Self Cite

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

“…In this section we extend the bound O(1/r 2 ) from [10] First, we show that, for the hypercube K = [−1, 1] n , we still have E…”

Section: Special Convex Bodiesmentioning

confidence: 80%

Section: Our Contributionmentioning

confidence: 99%

“…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%

mentioning

confidence: 97%

mentioning

confidence: 97%

See 3 more Smart Citations

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

Slot

2020

Math. Program.

Self Cite

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

Klerk

Association for Women in Mathematics Series

2019

Self Cite

The generalized problem of moments is a conic linear optimization problem over the convex cone of positive Borel measures with given support. It has a large variety of applications, including global optimization of polynomials and rational functions, options pricing in finance, constructing quadrature schemes for numerical integration, and distributionally robust optimization. A usual solution approach, due to J.B. Lasserre, is to approximate the convex cone of positive Borel measures by finite dimensional outer and inner conic approximations. We will review some results on these approximations, with a special focus on the convergence rate of the hierarchies of upper and lower bounds for the general problem of moments that are obtained from these inner and outer approximations.

Sum-of-Squares Hierarchies for Binary Polynomial Optimization

Slot

Lecture Notes in Computer Science

2021

Self Cite

We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial f over the boolean hypercube B n = {0, 1} n . This hierarchy provides for each integer r ∈ N a lower bound f (r ) on the minimum f min of f , given by the largest scalar λ for which the polynomial f − λ is a sum-of-squares on B n with degree at most 2r . We analyze the quality of these bounds by estimating the worstcase error f min − f (r ) in terms of the least roots of the Krawtchouk polynomials. As a consequence, for fixed t ∈ [0, 1/2], we can show that this worst-case error in the regime r ≈ t • n is of the order 1/2 − √ t(1 − t) as n tends to ∞. Our proof combines classical Fourier analysis on B n with the polynomial kernel technique and existing results on the extremal roots of Krawtchouk polynomials. This link to roots of orthogonal polynomials relies on a connection between the hierarchy of lower bounds f (r ) and another hierarchy of upper bounds f (r ) , for which we are also able to establish the same error analysis. Our analysis extends to the minimization of a polynomial over the q-ary cube (Z/qZ) n . Furthermore, our results apply to the setting of matrix-valued polynomials.