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Qualitative Properties of k-Center Problems

Author

Listed:
  • Vo Si Trong Long

    (University of Science
    Vietnam National University)

  • Nguyen Mau Nam

    (Portland State University)

  • Jacob Sharkansky

    (Portland State University)

  • Nguyen Dong Yen

    (Vietnam Academy of Science and Technology)

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.

Suggested Citation

  • Vo Si Trong Long & Nguyen Mau Nam & Jacob Sharkansky & Nguyen Dong Yen, 2025. "Qualitative Properties of k-Center Problems," Journal of Optimization Theory and Applications, Springer, vol. 207(1), pages 1-23, October.
    • Handle: RePEc:spr:joptap:v:207:y:2025:i:1:d:10.1007_s10957-025-02753-x
    DOI: 10.1007/s10957-025-02753-x
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    References listed on IDEAS

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    1. Nguyen Thai An & Nguyen Mau Nam & Xiaolong Qin, 2020. "Solving k-center problems involving sets based on optimization techniques," Journal of Global Optimization, Springer, vol. 76(1), pages 189-209, January.
    2. Christian Günther & Christiane Tammer, 2016. "Relationships between constrained and unconstrained multi-objective optimization and application in location theory," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(2), pages 359-387, October.
    3. Amir Beck & Shoham Sabach, 2015. "Weiszfeld’s Method: Old and New Results," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 1-40, January.
    4. Hoang Tuy, 2016. "Convex Analysis and Global Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-31484-6, January.
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