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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) (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.Comment: 14 pages, 2 figure
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