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.
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Notes
This value of \(c_d\) can be easily derived from results in [9] (specifically from Theorem 4.1, Lemma 4.6 and its proof).
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We thank the associate editor and two anonymous referees for their comments which helped improve the presentation of the paper.
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de Klerk, E., Laurent, M., Sun, Z. et al. On the convergence rate of grid search for polynomial optimization over the simplex. Optim Lett 11, 597–608 (2017). https://doi.org/10.1007/s11590-016-1023-7
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DOI: https://doi.org/10.1007/s11590-016-1023-7