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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.

##### MSC:
 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
##### Software:
flexclust; SIGOA; SymDIRECT
Full Text:
##### References:
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