Markov inequalities, Dubiner distance, norming meshes and polynomial optimization on convex bodies

Abstract

We construct norming meshes for polynomial optimization by the classical Markov inequality on general convex bodies in R d , and by a tangential Markov inequality via an estimate of the Dubiner distance on smooth convex bodies. These allow to compute a (1−ε)-approximation to the minimum of any polynomial of degree not exceeding n by O (n/ √ ε) αd samples, with α = 2 in the general case, and α = 1 in the smooth case. Such constructions are based on three cornerstones of convex geometry, Bieberbach volume inequality and Leichtweiss inequality on the affine breadth eccentricity, and the Rolling Ball Theorem, respectively.

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

“…For the example (15) we can also give an explicit expression for the bounds f (d) and we will show that their convergence rate to f min,K is also in the order Ω(1/d 2 ), which shows that the analysis in [3] is tight.…”

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

“…For a general convex body K some constructions are proposed recently in [45] for suitable grid point sets (so-called meshed norming sets) X d (ǫ) ⊆ K where d ∈ N and ǫ > 0. Namely, whenever p has degree at most d, by minimizing p over X d (ǫ) one obtains an upper bound on the minimum of p over K satisfying…”

Approximation of the Constant in a Markov-Type Inequality on a Simplex Using Meta-Heuristics

“…They are often used in the error analysis of variational techniques, including the finite element method or the discontinuous Galerkin method used for solving partial differential equations (PDEs) [10]. Moreover, Markov's inequality can be found in constructions of polynomial meshes, as it is a representative sampling method, and is used as a tool for studying the uniform convergence of discrete least-squares polynomial approximations or for spectral methods for the solutions of PDEs [11][12][13]. Generally, finding exact values of optimal constants in a given compact set E in R N is considered particularly challenging.…”

“…F. Piazzon and M. Vianello [11] used the approximation theory notions of a polynomial mesh and the Dubiner distance in a compact set to determine error estimates for the total degree of polynomial optimization on Chebyshev grids of the hypercube. Additionally, F. Piazzon and M. Vianello [12] constructed norming meshes for polynomial optimization by using a classical Markov inequality on the general convex body in R N . A.…”