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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 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.

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Informasi Detail

Judul Seri

SIAM JOURNAL ON OPTIMIZATION; Vol.31 No.3 July-September 2021

No. Panggil

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Penerbit

: .,

Deskripsi Fisik

p. 2255-2306

Bahasa

English

ISBN/ISSN

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Klasifikasi

NONE

Tipe Isi

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Tipe Media

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Tipe Pembawa

-

Edisi

-

Subjek

NUMERICAL INTEGRATION
SUMS OF SQUARES
SEMIDEFINITE PROGRAMMING
POLYNOMIAL OPTIMIZATION
PUSHFORWARD MEASURE
UPPER BOUNDS HIERARCHY

Info Detail Spesifik

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Pernyataan Tanggungjawab

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

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