Minimizing Rational Functions: A Hierarchy of Approximations via Pushforward Measures

Jean-Bernard Lasserre, Victor Magron, Swann Marx, Olivier Zahm

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 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 on 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 the Lebesgue or Haar measure.