Distributionally robust optimization with polynomial densities: theory, models and algorithms

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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]:

mentioning

confidence: 67%

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

Klerk

Laurent

2020

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“…Application to the generalized problem of moments (GPM) and cubature rules. As shown in [7] results on the convergence analysis of the bounds E (r) (f ) have direct implications for the following generalized moment problem (GMP):…”

Section: Compact Sets Satisfying Assumptionmentioning

confidence: 99%

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

Slot

Laurent

2020

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

“…Similarly, the family of measures on Ω with SOS-densities (Ex. 1.7) discussed in de Klerk et al [10] also have a great modeling power with nice properties. 2πσ exp(− (ω − a) 2 2σ 2 ) dω, that is µ is a mixture of Gaussian probability measures with mean-deviation couple (a, σ) ∈ A.…”

Section: Introductionmentioning

confidence: 95%

“…The resulting ambiguity set has been already used in several contributions like e.g. [9,15,16,17,51]; however, as discussed in [10], the resulting ambiguity set might be too overly conservative (even in the case where A is the singleton {(m, σ)}). The methodology developed in this paper also applies with some ad-hoc modification; see §5.1.…”

Section: Introductionmentioning

confidence: 99%

Distributionally robust polynomial chance-constraints under mixture ambiguity sets

Lasserre

Weisser

2019

Given X ⊂ R n , ε ∈ (0, 1), a parametrized family of probability distributions (µa) a∈A on Ω ⊂ R p , we consider the feasible set X * ε ⊂ X associated with the distributionally robust chance-constraintwhere Ma is the set of all possibles mixtures of distributions µa, a ∈ A. For instance and typically, the family Ma is the set of all mixtures of Gaussian distributions on R with mean and standard deviation a = (a, σ) in some compact set A ⊂ R 2 . We provide a sequence of inner approximations X d ε = {x ∈ X : w d (x) < ε}, d ∈ N, where w d is a polynomial of degree d whose vector of coefficients is an optimal solution of a semidefinite program. The size of the latter increases with the degree d. We also obtain the strong and highly desirable asymptotic guarantee that λ(X * ε \ X d ε ) → 0 as d increases, where λ is the Lebesgue measure on X. Same results are also obtained for the more intricated case of distributionally robust "joint" chance-constraints.