### Abstract

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 language | English |
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Journal | Mathematical Programming |

DOIs | |

Publication status | Accepted/In press - Jan 2020 |

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**Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere.** / de Klerk, Etienne; Laurent, Monique.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere

AU - de Klerk, Etienne

AU - Laurent, Monique

PY - 2020/1

Y1 - 2020/1

N2 - 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.

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.

U2 - doi:10.1007/s10107-019-01465-1

DO - doi:10.1007/s10107-019-01465-1

M3 - Article

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

ER -