An elementary approach to polynomial optimization on polynomial meshes

A polynomial mesh on a multivariate compact set or manifold is a sequence of finite norming sets for polynomials whose norming constant is independent of degree. We apply the recently developed theory of polynomial meshes to an elementary discrete approach for polynomial optimization on nonstandard domains, providing a rigorous (over)estimate of the convergence rate. Examples include surface/solid subregions of sphere or torus, such as caps, lenses, lunes, and slices

Optimal polynomial meshes and Caratheodory-Tchakaloff submeshes on the sphere

Using the notion of Dubiner distance, we give an elementary proof of the fact that good covering point configurations on the 2-sphere are optimal polynomial meshes. From these we extract Caratheodory-Tchakaloff (CATCH) submeshes for compressed Least Squares fitting

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

The Complexity of Optimizing over a Simplex, Hypercube or Sphere: A Short Survey

We consider the computational complexity of optimizing various classes of continuous functions over a simplex, hypercube or sphere.These relatively simple optimization problems have many applications.We review known approximation results as well as negative (inapproximability) results from the recent literature.computational complexity;global optimization;linear and semidefinite programming;approximation algorithms