10,645 research outputs found
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
Energy bounds for codes and designs in Hamming spaces
We obtain universal bounds on the energy of codes and for designs in Hamming
spaces. Our bounds hold for a large class of potential functions, allow unified
treatment, and can be viewed as a generalization of the Levenshtein bounds for
maximal codes.Comment: 25 page
Worst-case examples for Lasserre's measure--based hierarchy for polynomial optimization on the hypercube
We study the convergence rate of a hierarchy of upper bounds for polynomial
optimization problems, proposed by Lasserre [SIAM J. Optim. 21(3) (2011), pp.
864-885], and a related hierarchy by De Klerk, Hess and Laurent [SIAM J. Optim.
27(1), (2017) pp. 347-367]. For polynomial optimization over the hypercube, we
show a refined convergence analysis for the first hierarchy. We also show lower
bounds on the convergence rate for both hierarchies on a class of examples.
These lower bounds match the upper bounds and thus establish the true rate of
convergence on these examples. Interestingly, these convergence rates are
determined by the distribution of extremal zeroes of certain families of
orthogonal polynomials.Comment: 17 pages, no figure
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