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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. (English) Zbl 07069294
Summary: 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.

65K05 Numerical mathematical programming methods
90C26 Nonconvex programming, global optimization
90C27 Combinatorial optimization
90C56 Derivative-free methods and methods using generalized derivatives
90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
05E05 Symmetric functions and generalizations
Full Text: DOI
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