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Searching approximate global optimal Heilbronn configurations of nine points in the unit square via GPGPU computing

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Abstract

In this paper we present a method of applying the GPGPU technology to compute the approximate optimal solution to the Heilbronn problem for nine points in the unit square, namely, points \(P_1,P_2,\ldots ,P_9\) in \([0,1]\times [0,1]\) so that the minimal area of triangles \(P_iP_jP_k\,(1\le i<j<k\le 9)\) gets the maximal value \(h_9(\Box )\). We construct nine rectangles with edge length 1 / 80 in the unit square which contain all optimal Heilbronn configurations up to possible rotation and reflection, and prove that \(\frac{9\sqrt{65}-55}{320}=0.054875999\cdots<h_9(\Box )<0.054878314\), the lower bound here comes from Comellas and Yebra’s paper.

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Acknowledgments

The authors would like to thank Prof. Dr. Marc Moreno Maza for his helpful corrections and suggestions. The authors also thank Prof. Dr. Ju Zhang and Chongqing Institute of Green and Intelligent Technology of the Chinese Academy of China for GPGPU support.

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Authors and Affiliations

  1. Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, China

    Liangyu Chen & Yaochen Xu

  2. Department of Mathematics, Shanghai University, Shanghai, China

    Zhenbing Zeng

Authors
  1. Liangyu Chen
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  2. Yaochen Xu
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  3. Zhenbing Zeng
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Corresponding author

Correspondence to Yaochen Xu.

Additional information

This research was supported by the Natural Science Foundation of China (Nos. 11471209, 61321064), and Fundamental Research Funds for the Central Universities (No. 78210152).

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Chen, L., Xu, Y. & Zeng, Z. Searching approximate global optimal Heilbronn configurations of nine points in the unit square via GPGPU computing. J Glob Optim 68, 147–167 (2017). https://doi.org/10.1007/s10898-016-0453-1

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  • DOI: https://doi.org/10.1007/s10898-016-0453-1

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