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
N1 - Funding Information:
We thank two anonymous referees for their useful remarks. This work has been supported by European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie Grant Agreement 813211 (POEMA).
Publisher Copyright:
© 2020, The Author(s).
PY - 2022/6
Y1 - 2022/6
N2 - 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 is an element of 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/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.
AB - 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 is an element of 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/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.
KW - polynomial optimization on sphere
KW - Lasserre hierarchy
KW - semidefinite programming
KW - generalized eigenvalue problem
U2 - 10.1007/s10107-019-01465-1
DO - 10.1007/s10107-019-01465-1
M3 - Article
VL - 193
SP - 665
EP - 685
JO - Mathematical Programming
JF - Mathematical Programming
SN - 0025-5610
IS - 2
ER -