Solving moment and polynomial optimization problems on Sobolev spaces

TL;DR

This paper develops a hierarchy of semidefinite approximations to solve moment and polynomial optimization problems on Sobolev spaces, enabling numerical solutions with convergence guarantees.

Contribution

It introduces a novel moment-sums of squares hierarchy tailored for Sobolev spaces, extending polynomial optimization techniques to infinite-dimensional settings.

Findings

Established outer and inner semidefinite approximations of Sobolev moment cones

Developed an infinite-dimensional hierarchy with global convergence guarantees

Demonstrated the approach's effectiveness through numerical examples

Abstract

Using standard tools of harmonic analysis, we state and solve the problem of moments for non-negative measures supported on the unit ball of a Sobolev space of multivariate periodic trigonometric functions. We describe outer and inner semidefinite approximations of the cone of Sobolev moments. They are the basic components of an infinite-dimensional moment-sums of squares hierarchy, allowing to numerically solve non-convex polynomial optimization problems on infinite-dimensional Sobolev spaces with global convergence guarantees