Qualitative Properties of k-Center Problems

Vo Si Trong Long (), Nguyen Mau Nam (), Jacob Sharkansky () and Nguyen Dong Yen ()
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Vo Si Trong Long: University of Science
Nguyen Mau Nam: Portland State University
Jacob Sharkansky: Portland State University
Nguyen Dong Yen: Vietnam Academy of Science and Technology

Journal of Optimization Theory and Applications, 2025, vol. 207, issue 1, No 1, 23 pages

Abstract: Abstract In this paper, we study generalized versions of the k-center problem, which involves finding $$k$$ k circles of the smallest possible equal radius that cover a finite set of points in the plane. By utilizing the Minkowski gauge function, we extend this problem to generalized balls induced by various convex sets in finite dimensions, rather than limiting it to circles in the plane. First, we establish several fundamental properties of the global optimal solutions to this problem. We then introduce the notion of local optimal solutions and provide a sufficient condition for their existence. We also provide several illustrative examples to clarify the proposed problems.

Keywords: k-center problem; Global optimal solution; Local optimal solution; DC programming; Minkowski gauge function; 49J52; 68Q25; 90C26; 90C90; 90C31; 90C35 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-025-02753-x

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