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

We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre (SIAM J Optim 21(3):864–885, 2011), 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...

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Published in:Mathematical programming Vol. 193; no. 2; pp. 665 - 685
Main Authors: de Klerk, Etienne, Laurent, Monique
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2022
Springer
Springer Nature B.V
Subjects:
Analysis
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Convergence
Distribution (Probability theory)
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Polynomials
Probability density functions
Theoretical
Upper bounds
Generalized eigenvalue problem
Semidefinite programming
90C30
Polynomial optimization on sphere
90C22
Lasserre hierarchy
90C26
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We study the convergence rate of a hierarchy of upper bounds for polynomial minimization problems, proposed by Lasserre (SIAM J Optim 21(3):864–885, 2011), 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 2 r with respect to the surface measure. We show that the rate of convergence is O ( 1 / r 2 ) and we give a class of polynomials of any positive degree for which this rate is tight. In addition, we explore the implications for the related rate of convergence for the generalized problem of moments on the sphere.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-019-01465-1