  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.
