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January  2017, 13(1): 147-153. doi: 10.3934/jimo.2016009

## An efficient cutting plane algorithm for the smallest enclosing circle problem

 1 VC/VR Lab and Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan 610066, China 2 Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan 610066, China 3 Neijiang Vocational Technical College, Neijiang, Sichuan 641100, China

* Corresponding author

Received  April 2015 Revised  December 2015 Published  March 2016

Fund Project: The first author is supported by National Natural Science Foundation of China No.11126336 and No.11201324, New Teachers’ Fund for Doctor Stations, Ministry of Education No.20115134120001, Fok Ying Tuny Education Foundation No.141114, Youth fund of Sichuan province No.2013JQ0027.

In this paper, we consider the problem of computing the smallest enclosing circle. An efficient cutting plane algorithm is derived. It is based on finding the valid cut and reducing the problem into solving a series of linear programs. The numerical performance of this algorithm outperforms other existing algorithms in our computational experiments.

Citation: Yi Jiang, Chuan Luo, Shenggui Ling. An efficient cutting plane algorithm for the smallest enclosing circle problem. Journal of Industrial and Management Optimization, 2017, 13 (1) : 147-153. doi: 10.3934/jimo.2016009
##### References:
 [1] M. Berg, Computational Geometry: Algorithms and Applicaions, Springer, New York, 1997. doi: 10.1007/978-3-662-03427-9. [2] P. Chrystal, On the problem to construct the minimum circle enclosing n given points in a plan, in Proceedings of the Edinburgh Mathematical Society, Tird Meeting, 1885, 30. [3] J. Eliosoff and R. Unger, Minimal spanning circle of a set of points, Computer Science: Computational Geometry Project, School of Computer Science, McGill University, 1998,308– 507. [4] J. Elzinga and D. Hearn, The minimum covering sphere problem, Management Sci., 19 (1972), 96-104.  doi: 10.1287/mnsc.19.1.96. [5] B. Gärtner, Fast and robust smallest enclosing balls, in Algorithms-ESA' 99: 7th Annual European Symposium, Proceedings (eds. J. Nestril), Lecture Notes in Computer Science, 1643, Springer-Verlag, 1999,325–338. [6] D. W. Hearn and J. Vijan, Efficient algorithms for the minimum circle problem, Oper. Res., 30 (1982), 777-795.  doi: 10.1287/opre.30.4.777. [7] M. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programming, Linear Albebra and its Applications, 284 (1998), 193-228.  doi: 10.1016/S0024-3795(98)10032-0. [8] J. Sturm, Using SeDuMi 1. 02, a MATLAB toolbox for optimization over symmetric cones Optim. Methods Softw., 11/12 (1999), 625-653. doi: 10.1080/10556789908805766. [9] Y. Tian, S. C. Fang, Z. B. Deng and W. X. Xing, Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming, Journal of Industrial and Management Optimization, 9 (2013), 703-721.  doi: 10.3934/jimo.2013.9.703. [10] S. Y. Wang, Y. J. Liu and Y. Jiang, A majorized penalty approach to inverse linear second order cone programming problems, Journal of Industrial and Management Optimization, 10 (2014), 965-976.  doi: 10.3934/jimo.2014.10.965. [11] E. Welzl, Smalleat enclosing disks (balls and ellipsoids), in New Results and New Trends in Computer Science, Springer-Verlag, 1991,359–370. doi: 10.1007/BFb0038202. [12] S. Xu, R. Freund and J. Sun, Solution methodologies for the smallest enclosing circle problem, Comput. Optim. Appl., 25 (2003), 283-292.  doi: 10.1023/A:1022977709811. [13] Y. Zhang, Y. Jiang, L. Zhang and J. Zhang, A perturbation approach for an inverse linear second-order cone programming, Journal of Industrial and Management Optimization, 9 (2013), 171-189.  doi: 10.3934/jimo.2013.9.171.

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##### References:
 [1] M. Berg, Computational Geometry: Algorithms and Applicaions, Springer, New York, 1997. doi: 10.1007/978-3-662-03427-9. [2] P. Chrystal, On the problem to construct the minimum circle enclosing n given points in a plan, in Proceedings of the Edinburgh Mathematical Society, Tird Meeting, 1885, 30. [3] J. Eliosoff and R. Unger, Minimal spanning circle of a set of points, Computer Science: Computational Geometry Project, School of Computer Science, McGill University, 1998,308– 507. [4] J. Elzinga and D. Hearn, The minimum covering sphere problem, Management Sci., 19 (1972), 96-104.  doi: 10.1287/mnsc.19.1.96. [5] B. Gärtner, Fast and robust smallest enclosing balls, in Algorithms-ESA' 99: 7th Annual European Symposium, Proceedings (eds. J. Nestril), Lecture Notes in Computer Science, 1643, Springer-Verlag, 1999,325–338. [6] D. W. Hearn and J. Vijan, Efficient algorithms for the minimum circle problem, Oper. Res., 30 (1982), 777-795.  doi: 10.1287/opre.30.4.777. [7] M. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programming, Linear Albebra and its Applications, 284 (1998), 193-228.  doi: 10.1016/S0024-3795(98)10032-0. [8] J. Sturm, Using SeDuMi 1. 02, a MATLAB toolbox for optimization over symmetric cones Optim. Methods Softw., 11/12 (1999), 625-653. doi: 10.1080/10556789908805766. [9] Y. Tian, S. C. Fang, Z. B. Deng and W. X. Xing, Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming, Journal of Industrial and Management Optimization, 9 (2013), 703-721.  doi: 10.3934/jimo.2013.9.703. [10] S. Y. Wang, Y. J. Liu and Y. Jiang, A majorized penalty approach to inverse linear second order cone programming problems, Journal of Industrial and Management Optimization, 10 (2014), 965-976.  doi: 10.3934/jimo.2014.10.965. [11] E. Welzl, Smalleat enclosing disks (balls and ellipsoids), in New Results and New Trends in Computer Science, Springer-Verlag, 1991,359–370. doi: 10.1007/BFb0038202. [12] S. Xu, R. Freund and J. Sun, Solution methodologies for the smallest enclosing circle problem, Comput. Optim. Appl., 25 (2003), 283-292.  doi: 10.1023/A:1022977709811. [13] Y. Zhang, Y. Jiang, L. Zhang and J. Zhang, A perturbation approach for an inverse linear second-order cone programming, Journal of Industrial and Management Optimization, 9 (2013), 171-189.  doi: 10.3934/jimo.2013.9.171.
The smallest enclosing circle for eight circles by the cutting plane method
Computational results for 12800 circles by the cutting plane method
 k R (x, y) Time 1 20.9058731071536 (0.545051135217305, 1.44782921739902) 3122 2 20.9105991582134 (0.299192662259897, 1.31150575665027) 3666 3 20.9125332765331 (0.298922131451905, 1.31135826866618) 4291 4 20.9125332765331 (0.298922131451905, 1.31135826866618) 4934
 k R (x, y) Time 1 20.9058731071536 (0.545051135217305, 1.44782921739902) 3122 2 20.9105991582134 (0.299192662259897, 1.31150575665027) 3666 3 20.9125332765331 (0.298922131451905, 1.31135826866618) 4291 4 20.9125332765331 (0.298922131451905, 1.31135826866618) 4934
The number of iterations on the cutting plane method
 m Average Maximum Minimum 50 3.14 5 2 200 3.08 4 3 800 3.22 4 3 3200 3.16 5 3 12800 3.46 7 3
 m Average Maximum Minimum 50 3.14 5 2 200 3.08 4 3 800 3.22 4 3 3200 3.16 5 3 12800 3.46 7 3
Objective function value
 Problem Obj Value m SOCP QP Algorithm 1 50 11.2096459 11.2096469 11.2096459 200 14.3832518 14.3832526 14.3832516 800 16.9222886 16.9222890 16.9222882 3200 19.1035735 19.1035733 19.1035728 12800 20.9684117 Out of Memory 20.9684103
 Problem Obj Value m SOCP QP Algorithm 1 50 11.2096459 11.2096469 11.2096459 200 14.3832518 14.3832526 14.3832516 800 16.9222886 16.9222890 16.9222882 3200 19.1035735 19.1035733 19.1035728 12800 20.9684117 Out of Memory 20.9684103
Average CPU time of three methods
 Problem Time m SOCP QP Algorithm 1 50 82.2 88.4 79.8 200 115.0 180.2 103.4 800 257.4 1859.4 254.6 3200 1041.6 34883.9 973.1 12800 8555.7 Out of Memory 5024.6
 Problem Time m SOCP QP Algorithm 1 50 82.2 88.4 79.8 200 115.0 180.2 103.4 800 257.4 1859.4 254.6 3200 1041.6 34883.9 973.1 12800 8555.7 Out of Memory 5024.6
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