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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Vydané v:	Mathematical programming Ročník 193; číslo 2; s. 665 - 685
Hlavní autori:	de Klerk, Etienne, Laurent, Monique
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2022 Springer Springer Nature B.V
Predmet:	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
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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):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.
AbstractList	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 $$r \in {\mathbb {N}}$$ 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 $$O(1/r^2)$$ 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. 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 2r with respect to the surface measure. We show that the rate of convergence is O(1/r2) 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. 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. 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 [Formula omitted] 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 rate of convergence is [Formula omitted] 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.
Audience	Academic
Author	Laurent, Monique de Klerk, Etienne
Author_xml	– sequence: 1 givenname: Etienne orcidid: 0000-0003-3377-0063 surname: de Klerk fullname: de Klerk, Etienne email: E.deKlerk@uvt.nl organization: Tilburg University – sequence: 2 givenname: Monique surname: Laurent fullname: Laurent, Monique organization: Tilburg University, Centrum Wiskunde & Informatica (CWI)
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Cites_doi	10.1007/978-1-4614-6660-4 10.1137/100806990 10.1007/s10107-003-0387-5 10.1007/10997703_7 10.1007/s11590-019-01416-x 10.1137/S1052623400366802 10.1016/j.jat.2012.05.014 10.4153/CJM-1965-053-6 10.1017/CBO9780511565717 10.1007/s10107-011-0499-2 10.1287/moor.2017.0906 10.1287/moor.1060.0194 10.1016/j.jat.2009.11.006 10.1007/s10107-016-1043-1 10.1007/s10107-019-01429-5 10.1090/conm/253/03936 10.1007/s10107-020-01468-3 10.1007/978-3-030-21170-7_1
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Snippet	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... 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...
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SubjectTerms	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
Title	Convergence analysis of a Lasserre hierarchy of upper bounds for polynomial minimization on the sphere
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