research-article

Computing the global optimum of a multivariate polynomial over the reals

Author:

Mohab Safey El DinAuthors Info & Claims

ISSAC '08: Proceedings of the twenty-first international symposium on Symbolic and algebraic computation

Pages 71 - 78

https://doi.org/10.1145/1390768.1390781

Published: 20 July 2008 Publication History

Abstract

Let f be a polynomial in Q[X₁, ..., X_n] of degree D. We provide an efficient algorithm in practice to compute the global supremum sup_{x∈ Rⁿ} f(x) of f (or its infimum inf_{{x∈ Rⁿ}}f(x)). The complexity of our method is bounded by D^O(n)}. In a probabilistic model, a more precise result yields a complexity bounded by O(n⁷D⁴ⁿ) arithmetic operations in Q. Our implementation is more efficient by several orders of magnitude than previous ones based on quantifier elimination. Sometimes, it can tackle problems that numerical techniques do not reach. Our algorithm is based on the computation of generalized critical values of the mapping x-> f(x), i.e. the set of points {c∈ C mid exists (x_ll)_ll∈ N}⊂ Cⁿ ;f(x_ll)-> c, ;||x_ll||||d_{x_ll} f||-> 0 { when }ll-> ∞}. We prove that the global optimum of f lies in its set of generalized critical values and provide an efficient way of deciding which value is the global optimum.

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Wang CYang ZZhi L(2019)Global Optimization of Polynomials over Real Algebraic SetsJournal of Systems Science and Complexity10.1007/s11424-019-8351-532:1(158-184)Online publication date: 14-Feb-2019
https://doi.org/10.1007/s11424-019-8351-5
Dias LTanabé STibăźR M(2017)Toward Effective Detection of the Bifurcation Locus of Real Polynomial MapsFoundations of Computational Mathematics10.1007/s10208-016-9303-217:3(837-849)Online publication date: 1-Jun-2017
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Safey El DIn MSpaenlehauer PAbramov SZima EGao X(2016)Critical Point Computations on Smooth VarietiesProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930929(183-190)Online publication date: 20-Jul-2016
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Computing the global optimum of a multivariate polynomial over the reals

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cover image ACM Conferences

ISSAC '08: Proceedings of the twenty-first international symposium on Symbolic and algebraic computation

July 2008

348 pages

ISBN:9781595939043

DOI:10.1145/1390768

General Chair:
J. Rafael Sendra
Universidad de Alcalá, Spain
,
Program Chair:
Laureano Gonzalez-Vega
Universidad de Cantabria, Spain

Copyright © 2008 ACM.

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Published: 20 July 2008

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July 20 - 23, 2008

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Cited By

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Dias LTanabé STibăźR M(2017)Toward Effective Detection of the Bifurcation Locus of Real Polynomial MapsFoundations of Computational Mathematics10.1007/s10208-016-9303-217:3(837-849)Online publication date: 1-Jun-2017
https://dl.acm.org/doi/10.1007/s10208-016-9303-2
Safey El DIn MSpaenlehauer PAbramov SZima EGao X(2016)Critical Point Computations on Smooth VarietiesProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930929(183-190)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930929
Faugère JSpaenlehauer PSvartz JAbramov SZima EGao X(2016)Computing Small Certificates of Inconsistency of Quadratic Fewnomial SystemsProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930927(223-230)Online publication date: 20-Jul-2016
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Guo FSafey El Din MWang CZhi LRobertz DLinton SYokoyama K(2015)Optimizing a Parametric Linear Function over a Non-compact Real Algebraic VarietyProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756666(205-212)Online publication date: 24-Jun-2015
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Greuet ASafey El Din M(2014)Probabilistic Algorithm for Polynomial Optimization over a Real Algebraic SetSIAM Journal on Optimization10.1137/13093130824:3(1313-1343)Online publication date: 1-Jan-2014
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Guo QSafey EI Din MZhi LMonagan MCooperman GGiesbrecht M(2013)Computing rational solutions of linear matrix inequalitiesProceedings of the 38th International Symposium on Symbolic and Algebraic Computation10.1145/2465506.2465949(197-204)Online publication date: 26-Jun-2013
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