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

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Abstract

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 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 Theta(1/r^2), and explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.
Original languageEnglish
JournalMathematical Programming
DOIs
Publication statusAccepted/In press - Jan 2020

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Convergence Analysis
Rate of Convergence
Polynomials
Upper bound
Polynomial
Related rates
Sum of squares
Expected Value
Minimization Problem
Probability density function
Probability distributions
Probability Distribution
Moment
Unit
Hierarchy

Cite this

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title = "Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere",
abstract = "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 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 Theta(1/r^2), and explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.",
author = "{de Klerk}, Etienne and Monique Laurent",
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AB - 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 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 Theta(1/r^2), and explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.

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