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Optimization Letters

, Volume 11, Issue 3, pp 597–608 | Cite as

On the convergence rate of grid search for polynomial optimization over the simplex

  • Etienne de Klerk
  • Monique Laurent
  • Zhao Sun
  • Juan C. Vera
Original Paper

Abstract

We consider the approximate minimization of a given polynomial on the standard simplex, obtained by taking the minimum value over all rational grid points with given denominator \({r} \in \mathbb {N}\). It was shown in De Klerk et al. (SIAM J Optim 25(3):1498–1514, 2015) that the accuracy of this approximation depends on r as \(O(1/r^2)\) if there exists a rational global minimizer. In this note we show that the rational minimizer condition is not necessary to obtain the \(O(1/r^2)\) bound.

Keywords

Polynomial optimization Grid search Convergence rate  Taylor’s theorem 

Notes

Acknowledgments

We thank the associate editor and two anonymous referees for their comments which helped improve the presentation of the paper.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Etienne de Klerk
    • 1
  • Monique Laurent
    • 2
  • Zhao Sun
    • 3
  • Juan C. Vera
    • 1
  1. 1.Tilburg UniversityTilburgNetherlands
  2. 2.Centrum Wiskunde and Informatica (CWI)Amsterdam and Tilburg University, CWIAmsterdamNetherlands
  3. 3.Canada Excellence Research Chair in “Data Science for Real-time Decision making”École Polytechnique de MontréalMontréalCanada

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