1,830 research outputs found
Subperiodic Dubiner distance, norming meshes and trigonometric polynomial optimization
We extend the notion of Dubiner distance from algebraic to trigonometric polynomials on subintervals of the period, and we obtain its explicit form by the Szego variant of Videnskii inequality. This allows to improve previous estimates for Chebyshev-like trigonometric norming meshes, and suggests a possible use of such meshes in the framework of multivariate polynomial optimization on regions defined by circular arcs
A note on total degree polynomial optimization by Chebyshev grids
Using the approximation theory notions of polynomial mesh and Dubiner distance in a compact set, we derive error estimates for total degree polynomial optimization on Chebyshev grids of the hypercub
On the complexity of Putinar's Positivstellensatz
We prove an upper bound on the degree complexity of Putinar's
Positivstellensatz. This bound is much worse than the one obtained previously
for Schm\"udgen's Positivstellensatz but it depends on the same parameters. As
a consequence, we get information about the convergence rate of Lasserre's
procedure for optimization of a polynomial subject to polynomial constraints
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
An Efficient Framework for Global Non-Convex Polynomial Optimization over the Hypercube
We present a novel efficient theoretical and numerical framework for solving
global non-convex polynomial optimization problems. We analytically demonstrate
that such problems can be efficiently reformulated using a non-linear objective
over a convex set; further, these reformulated problems possess no spurious
local minima (i.e., every local minimum is a global minimum). We introduce an
algorithm for solving these resulting problems using the augmented Lagrangian
and the method of Burer and Monteiro. We show through numerical experiments
that polynomial scaling in dimension and degree is achievable for computing the
optimal value and location of previously intractable global polynomial
optimization problems in high dimension
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