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

  • Rudolf ScitovskiEmail author
  • Kristian Sabo


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.


Globally optimal partition k-means Incremental algorithm DIRECT Multiple circles detection problem 

Mathematics Subject Classification

65K05 90C26 90C27 90C56 90C57 05E05 



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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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsijekOsijekCroatia

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