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Application of the DIRECT algorithm to searching for an optimal k-partition of the set \(\mathcal {A}\subset \mathbb {R}^n\) and its application to the multiple circle detection problem

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Abstract

In this paper, we propose an efficient method for searching for a globally optimal k-partition of the set \(\mathcal {A}\subset \mathbb {R}^n\). Due to the property of the DIRECT global optimization algorithm to usually quickly arrive close to a point of global minimum, after which it slowly attains the desired accuracy, the proposed method uses the well-known k-means algorithm with a initial approximation chosen on the basis of only a few iterations of the DIRECT algorithm. In case of searching for an optimal k-partition of spherical clusters, the method is not worse than other known methods, but in case of solving the multiple circle detection problem, the proposed method shows remarkable superiority.

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Notes

  1. All evaluations were done on the basis of our own Mathematica-modules freely available at: https://www.mathos.unios.hr/images/homepages/scitowsk/GOPart.rar, and were performed on the computer with a 2.90 GHz Intel(R) Core(TM)i7-75000 CPU with 16GB of RAM.

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Acknowledgements

The author would like to thank the referees and the journal editors for their careful reading of the paper and insightful comments that helped us improve the paper. Especially, the author would like to thank Mrs. Katarina Moržan for significantly improving the use of English in the paper. This work was supported by the Croatian Science Foundation through research Grants IP-2016-06-6545 and IP-2016-06-8350

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  1. Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, 31 000, Osijek, Croatia

    Rudolf Scitovski & Kristian Sabo

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  1. Rudolf Scitovski
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  2. Kristian Sabo
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Correspondence to Rudolf Scitovski.

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Scitovski, R., Sabo, K. Application of the DIRECT algorithm to searching for an optimal k-partition of the set \(\mathcal {A}\subset \mathbb {R}^n\) and its application to the multiple circle detection problem. J Glob Optim 74, 63–77 (2019). https://doi.org/10.1007/s10898-019-00743-8

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