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Minimizing rational functions: a hierarchy of approximations via pushforward measures

Jean-Bernard Lasserre 1, 2 Victor Magron 1, 2 Swann Marx 3 Olivier Zahm 4 
2 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes
4 AIRSEA - Mathematics and computing applied to oceanic and atmospheric flows
Inria Grenoble - Rhône-Alpes, UGA - Université Grenoble Alpes, LJK - Laboratoire Jean Kuntzmann, Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology
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.
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Submitted on : Friday, December 11, 2020 - 8:20:19 AM
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Jean-Bernard Lasserre, Victor Magron, Swann Marx, Olivier Zahm. Minimizing rational functions: a hierarchy of approximations via pushforward measures. SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2021, 31 (3), pp.2285-2306. ⟨10.1137/20M138541X⟩. ⟨hal-03053386⟩



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