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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References
Alonso, J., Martini, H., Spirova, M.: Minimal enclosing discs, circumcircles, and circumcenters in normed planes. Comput. Geom. 45, 258–274 (2012)
An, L.T.H., Tao, P.D.: DC programming and DCA: thirty years of developments. Math. Prog. 169, 5–64 (2018)
An, N.T., Giles, D., Nam, N.M., Rector, R.B.: The log-exponential smoothing technique and Nesterov’s accelerated gradient method for generalized Sylvester problems. J. Optim. Theory Appl. 168, 559–583 (2016)
Artacho, F.J.A., Vuong, P.T.: The boosted DC algorithm for nonsmooth functions. arXiv:1812.06070 (2018)
Brimberg, J., Wesolowsky, G.O.: Locating facilities by minimax relative to closest points of demand areas. Comput. Oper. Res. 29, 625–636 (2002)
Cheng, C., Hu, X., Martin, C.: On the smallest enclosing balls. Commun. Inf. Syst. 6, 137–160 (2006)
Das, P., Chakraborti, N.R., Chaudhuri, Pk: Spherical minimax location problem. Comput. Optim. Appl. 18, 311–326 (2001)
Drager, L., Lee, J., Martin, C.: On the geometry of the smallest circle enclosing a finite set of points. J. Frankl. Inst. 344, 929–940 (2007)
Drezner, Z.: The \(p\)-center problem—heuristic and optimal algorithms. J. Oper. Res. Soc. 35, 741–7488 (1984)
Drezner, Z., Wesolowsky, G.O.: Minimax and maximin facility location problems on a sphere. Naval Res. Logist. 30, 305–312 (1983)
Elshaikh, A., Salhi, S., Brimberg, J., Mladenovic, N., Callaghan, B., Nagy, G.: Adaptive perturbation based heuristics: an application for the continuous \(p\)-centre problem. Comput. Oper. Res. 75, 1–11 (2016)
Fischer, k, Gärtner, B.: The smallest enclosing ball of balls: combinatorial structure and algorithms. Int. J. Comput. Geom. Appl. 14, 341–378 (2004)
Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoret. Comput. Sci. 38, 293–306 (1985)
Hakimi, S.: Optimum location of switching centers and the absolute centers and medians of a graph. Oper. Res. 12, 450–459 (1964)
Hakimi, S.: Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Oper. Res. 13, 462–475 (1965)
Hazewinkel, M. (ed.): Fermat–Torricelli Problem. Encyclopedia of Mathematics. Springer, Berlin (2001)
Hearn, D.W., Vijay, J.: Efficient algorithms for the (weighted) minimum circle problem. Oper. Res. 30, 777–795 (1981)
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Fundamentals. Springer, Berlin (1993)
Hunter, D.R., Lange, k: Tutorial on MM algorithms. Am. Stat. 58, 30–37 (2004)
Iyigun, C., Ben-Israel, A.: A generalized Weiszfeld method for the multi-facility location problem. Oper. Res. Lett. 38, 207–214 (2010)
Jahn, T., Kupitz, Y.S., Martini, H., Richter, C.: Minsum location extended to gauges and to convex sets. J. Optim. Theory Appl. 166, 711–746 (2015)
Lange, k, Hunter, D.R., Yang, I.: Optimization transfer using surrogate objective functions. J. Comput. Graph. Stat. 9, 1–59 (2000). (with discussion)
Mordukhovich, B.S., Nam, N.M.: Applications of variational analysis to a generalized Fermat–Torricelli problem. J. Optim. Theory Appl. 148, 431–454 (2011)
Mordukhovich, B.S., Nam, N.M., Villalobos, C.: The smallest enclosing ball problem and the smallest intersecting ball problem: existence and uniqueness of optimal solutions. Optim. Lett. 7, 839–853 (2013)
Mordukhovich, B.S., Nam, N.M.: An Easy Path to Convex Analysis and Applications. Morgan & Claypool Publishers, San Rafael (2014)
Nam, N.M., An, N.T., Salinas, J.: Applications of convex analysis to the smallest intersecting ball problem. J. Convex Anal. 19, 497–518 (2012)
Nam, N.M., Hoang, N.: A generalized Sylvester problem and a generalized Fermat–Torricelli problem. J. Convex Anal. 20, 669–687 (2013)
Nam, N.M., Rector, R.B., Giles, D.: Minimizing differences of convex functions with applications to facility location and clustering. J. Optim. Theory Appl. 173, 255–278 (2017)
Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Program. 103, 127–152 (2005)
Nickel, S., Puerto, J., Rodriguez-Chia, A.M.: An approach to location models involving sets as existing facilities. Math. Oper. Res. 28, 693–715 (2003)
Nielsen, F., Nock, R.: Approximating smallest enclosing balls with applications to machine learning. Int. J. Comput. Geom. Appl. 19, 389–414 (2009)
Puerto, J., Rodriguez-Chia, A.M.: On the structure of the solution set for the single facility location problem with average distances. Math. Program., Ser. A 128, 373–401 (2011)
Sylvester, J.J.: A question in the geometry of situation. Q. J. Math. 1, 79 (1857)
Tao, P.D., An, L.T.H.: Convex analysis approach to D.C. programming: theory, algorithms and applications. Acta Math. Vietnam. 22, 289–355 (1997)
Tao, P.D., An, L.T.H.: A D.C. optimization algorithm for solving the trust-region subproblem. SIAM J. Optim. 8, 476–505 (1998)
Welzl, E.: Smallest enclosing disks (balls and ellipsoids). In: Maurer, H. (ed.) Lecture Notes in Computer Science, vol. 555, pp. 359–370. Springer, Berlin (1991)
Yildirim, E.A.: On the minimum volume covering ellipsoid of ellipsoids. SIAM J. Optim. 17, 621–641 (2006)
Yildirim, E.A.: Two algorithms for the minimum enclosing ball problem. SIAM J. Optim. 19, 1368–1391 (2008)
Zhai, X.: Two Problems in Convex Conic Optimization. Master’s thesis, National University of Singapore (2007)
Zhou, G., Toh, K.C., Sun, J.: Efficient algorithms for the smallest enclosing ball problem. Comput. Optim. Appl. 30, 147–160 (2005)
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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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
Keywords
- k-center problem
- Multifacility location problem
- Majorization-minimization principle
- Difference of convex functions