Minimizing Rational Functions: A Hierarchy of Approximations via Pushforward Measures

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...

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Bibliographic Details
Published in:SIAM journal on optimization 2021-01, Vol.31 (3), p.2285-2306
Main Authors: Lasserre, Jean Bernard, Magron, Victor, Marx, Swann, Zahm, Olivier
Format: Article
Language:English
Subjects:
Mathematics
Optimization and Control
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Summary: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.
ISSN:1052-6234
1095-7189
DOI:10.1137/20M138541X