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 SunEmail author
  • Juan C. Vera
Original Paper


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.


Polynomial optimization Grid search Convergence rate  Taylor’s theorem 



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
    Email author
  • 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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