Minimizing rational functions: a hierarchy of approximations via pushforward measures

TL;DR

This paper introduces a new hierarchy of semidefinite programs based on pushforward measures to efficiently approximate the minimum of high-dimensional rational functions, improving over traditional methods.

Contribution

It proposes a novel approach using pushforward measures within Lasserre's hierarchy to reduce dimensionality and enhance approximation accuracy for rational function minimization.

Findings

Hierarchy converges to the true minimum from above

Effective for single rational functions via generalized eigenvalue problems

Numerical results demonstrate the approach's potential

Abstract

This paper is concerned with minimizing a sum of rational functions over a compact set of high-dimension. Our approach relies on the second Lasserre's hierarchy (also known as the upper bounds hierarchy) formulated on the pushforward measure in order to work in a space of smaller dimension. We show that in the general case the minimum can be approximated as closely as desired from above with a hierarchy of semidefinite programs problems or, in the particular case of a single fraction, with a hierarchy of generalized eigenvalue problems. We numerically illustrate the potential of using the pushforward measure rather than the standard upper bounds hierarchy. In our opinion, this potential should be a strong incentive to investigate a related challenging problem interesting in its own; namely integrating an arbitrary power of a given polynomial on a simple set (e.g., unit box or unit sphere) with respect to Lebesgue or Haar measure.