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Solving k-center problems involving sets based on optimization techniques

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Abstract

The continuous k-center problem aims at finding k balls with the smallest radius to cover a finite number of given points in \(\mathbb {R}^n\). In this paper, we propose and study the following generalized version of the k-center problem: Given a finite number of nonempty closed convex sets in \(\mathbb {R}^n\), find k balls with the smallest radius such that their union intersects all of the sets. Because of its nonsmoothness and nonconvexity, this problem is very challenging. Based on nonsmooth optimization techniques, we first derive some qualitative properties of the problem and then propose new algorithms to solve the problem. Numerical experiments are also provided to show the effectiveness of the proposed algorithms.

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  1. https://en.wikipedia.org/wiki/List_of_United_States_cities_by_population.

  2. https://www.mathworks.com/help/map/pcarree.html.

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Acknowledgements

The authors are very grateful to the two anonymous referees for taking their valuable time to read and give us invaluable comments that allowed us to improve the presentation and the content of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Hangzhou Normal University, Hangzhou, China

    Xiaolong Qin

  2. Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Nguyen Thai An & Xiaolong Qin

  3. Institute of Research and Development, Duy Tan University, Da Nang, Vietnam

    Nguyen Thai An

  4. Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR, 97207, USA

    Nguyen Mau Nam

  5. Department of Mathematics, Zhejiang Normal University, Jinhua, China

    Xiaolong Qin

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  1. Nguyen Thai An
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  2. Nguyen Mau Nam
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Correspondence to Xiaolong Qin.

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This article was supported by the National Natural Science Foundation of China under Grant No.11401152.

N. T. An and X. Qin: Research of the first author was supported by the National Natural Science Foundation of China under Grant No.11950410503, the China Postdoctoral Science Foundation under Grant No.2017M622991 and the Vietnam National Foundation for Science and Technology Development under Grant No.101.01-2017.325. N. M. Nam: Research of the second author was partly supported by the National Science Foundation under Grant DMS-1716057.

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An, N.T., Nam, N.M. & Qin, X. Solving k-center problems involving sets based on optimization techniques. J Glob Optim 76, 189–209 (2020). https://doi.org/10.1007/s10898-019-00834-6

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  • DOI: https://doi.org/10.1007/s10898-019-00834-6

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