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An Efficient Inexact Newton-CG Algorithm for the Smallest Enclosing Ball Problem of Large Dimensions

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Abstract

In this paper, we consider the problem of computing the smallest enclosing ball (SEB) of a set of m balls in \({\mathbb {R}}^n\), where the product mn is large. We first approximate the non-differentiable SEB problem by its log-exponential aggregation function and then propose a computationally efficient inexact Newton-CG algorithm for the smoothing approximation problem by exploiting its special (approximate) sparsity structure. The key difference between the proposed inexact Newton-CG algorithm and the classical Newton-CG algorithm is that the gradient and the Hessian-vector product are inexactly computed in the proposed algorithm, which makes it capable of solving the large-scale SEB problem. We give an adaptive criterion of inexactly computing the gradient/Hessian and establish global convergence of the proposed algorithm. We illustrate the efficiency of the proposed algorithm by using the classical Newton-CG algorithm as well as the algorithm from Zhou et al. (Comput Optim Appl 30:147–160, 2005) as benchmarks.

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Notes

  1. The terms in square brackets in (3.20) are constants in the inner CG iteration, since they are not related to the variable \(\tilde{d}\).

References

  1. Zhou, G., Toh, K.C., Sun, J.: Efficient algorithms for the smallest enclosing ball problem. Comput. Optim. Appl. 30, 147–160 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. ReVelle, C.S., Eiselt, H.A.: Location analysis: a synthesis and survey. Eur. J. Oper. Res. 165, 1–19 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burges, C.J.C.: A tutorial on support vector machines for pattern recognition. Data Min. Knowl. Discov. 2, 121–167 (1998)

    Article  Google Scholar 

  4. Nielsen, F., Nock, R.: Approximating smallest enclosing balls with applications to machine learning. Int. J. Comput. Geom. Appl. 19, 389–414 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chung, F.L., Deng, Z., Wang, S.: From minimum enclosing ball to fast fuzzy inference system training on large datasets. IEEE Trans. Fuzzy Sys. 17, 173–184 (2009)

    Article  Google Scholar 

  6. Chapelle, O., Vapnik, V., Bousquet, O., Mukherjee, S.: Choosing multiple parameters for support vector machines. Mach. Learn. 46, 131–159 (2002)

    Article  MATH  Google Scholar 

  7. Ben-Hur, A., Horn, D., Siegelmann, H.T., Vapnik, V.: Support vector clustering. J. Mach. Learn. Res. 2, 125–137 (2001)

    MATH  Google Scholar 

  8. Cervantes, J., Li, X., Yu, W., Li, K.: Support vector machine classification for large data sets via minimum enclosing ball clustering. Neurocomputing 71, 611–619 (2008)

    Article  Google Scholar 

  9. Bădoiu, M., Har-Peled, S., Indyk, P.: Approximate clustering via core-sets. Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pp. 250–257 (2002)

  10. Alon, N., Dar, S., Parnas, M., Ron, D.: Testing of clustering. SIAM Rev. 46, 285–308 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Deng, Z., Chung, F.L., Wang, S.: FRSDE: fast reduced set density estimator using minimal enclosing ball approximation. Pattern Recogn. 41, 1363–1372 (2008)

    Article  MATH  Google Scholar 

  12. Elzinga, D.J., Hearn, D.W.: The minimum covering sphere problem. Manag. Sci. 19, 96–104 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hearn, D.W., Vijan, J.: Efficient algorithms for the (weighted) minimum circle problem. Oper. Res. 30, 777–795 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Xu, S., Freund, R.M., Sun, J.: Solution methodologies for the smallest enclosing circle problem. Comput. Optim. Appl. 25, 283–292 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Megiddo, N.: Linear-time algorithms for linear programming in \({\mathbb{R}}^3\) and related problems. SIAM J. Comput. 12, 759–776 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  16. Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction, Texts and Monographs in Computer Science. Springer-Verlag, New York (1985)

    Book  MATH  Google Scholar 

  17. Shamos, M.I., Hoey, D.: Closest-point problems. Proceeding of the 16th Annual Symposium on Foundations of Computer Science, pp. 151–162 (1975)

  18. Larsson, T., Kalberg, L.: Fast and robust approximation of smallest enclosing balls in arbitrary dimensions. Eurographics Symp. Geom. Process. 32, 93–101 (2013)

    Google Scholar 

  19. Dyer, M.: A class of convex programs with applications to computational geometry. Proceedings of the 8th Annual Symposium on Computational Geometry, pp. 9–15 (1992)

  20. Welzl, E.: Smallest enclosing disks (balls and ellipsoids). Lecture Notes in Computer Science, vol. 555, pp. 359–370 (1991)

  21. Gärtner, B., Schönherr, S.: An efficient, exact, and generic quadratic programming solver for geometric optimization. Proceedings of the 16th Annual Symposium on Computational Geometry, pp. 110–118 (2000)

  22. Fischer, K., Gärtner, B.: The smallest enclosing ball of balls: combinatorial structure and algorithms. Int. J. Comput. Geom. Appl. 14, 341–378 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cheng, D., Hu, X., Martin, C.: On the smallest enclosing balls. Commun. Inf. Syst. 6, 137–160 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Boyd, S., Mutapcic, A.: Subgradient methods. Notes for EE364b, Standford University (Winter 2006–2007)

  25. Shor, N.Z.: Minimization Methods for Non-Differentiable Functions. Springer-Verlag, New York (1985)

    Book  MATH  Google Scholar 

  26. TüTüncü, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Prog. Ser. B. 95, 189–217 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xu, S.: Smoothing method for minimax problems. Comput. Optim. Appl. 20, 267–279 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pan, S.H., Li, X.S.: An efficient algorithm for the smallest enclosing ball problem in high dimensions. Appl. Math. Comput. 172, 49–61 (2006)

    MathSciNet  MATH  Google Scholar 

  29. Nesterov, Y.: Smooth minimization of non-smooth functions. Math. Prog. Ser. A. 103, 127–152 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Xiao, Y., Yu, B.: A truncated aggregate smoothing Newton method for minimax problems. Appl. Math. Comput. 216, 1868–1879 (2010)

    MathSciNet  MATH  Google Scholar 

  31. Polak, E., Royset, J.O., Womersley, R.S.: Algorithms with adaptive smoothing for finite minimax problems. J. Optim. Theory Appl. 119, 459–484 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Polak, E.: Optimization: Algorithms and Consistent Approximations. Springer-Verlag, New York (1997)

    Book  MATH  Google Scholar 

  33. Zang, I.: A smoothing-out technique for min–max optimization. Math. Prog. 19, 61–77 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  34. Li, J., Wu, Z., Long, Q.: A new objective penalty function approach for solving constrained minimax problems. J. Oper. Res. Soc. China 2, 93–108 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lü, Y.B., Wan, Z.P.: A smoothing method for solving bilevel multiobjective programming problems. J. Oper. Res. Soc. China 2, 511–525 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Chen, X.: Smoothing methods for nonsmooth, nonconvex minimization. Math. Program. 134, 71–99 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Li, X.S.: An aggregate function method for nonlinear programming. Sci. China Ser. A. 34, 1467–1473 (1991)

    MathSciNet  MATH  Google Scholar 

  38. Li, X.S., Fang, S.C.: On the entropic regularization method for solving min–max problems with applications. Math. Methods Oper. Res. 46, 119–130 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  39. Liu, D.C., Nocedal, J.: On the limited memory BFGS method for large scale optimization. Math. Prog. 45, 503–528 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  40. Chen, B., Harker, P.T.: A non-interior-point continuation method for linear complementarity problems. SIAM J. Matrix Anal. Appl. 14, 1168–1190 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kanzow, C.: Some noninterior continuation methods for linear complementarity problems. SIAM J. Matrix Anal. Appl. 17, 851–868 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  42. Smale, S.: Algorithms for solving equations. Proceedings of International Congress of Mathematicians (1987)

  43. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer-Verlag, New York (1999)

    Book  MATH  Google Scholar 

  44. Sun, W., Yuan, Y.X.: Optimization Theory and Methods: Nonlinear Programming. Springer Science+Business Media, New York (2006)

    MATH  Google Scholar 

  45. Arkin, E.M., Hassin, R., Levin, A.: Approximations for minimum and min–max vehicle routing problems. J. Algorithm 59, 1–18 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  46. Cherkaev, E., Cherkaev, A.: Minimax optimization problem of structural design. Comput. Struct. 86, 1426–1435 (2008)

    Article  MATH  Google Scholar 

  47. Al-Subaihi, I., Watson, G.A.: Fitting parametric curves and surfaces by \(l_{\infty }\) discrete regression. BIT Numer. Math. 45, 443–461 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  48. Liu, Y.F., Dai, Y.H., Luo, Z.Q.: Coordinated beamforming for MISO interference channel: complexity analysis and efficient algorithms. IEEE Trans. Signal Process. 59, 1142–1157 (2011)

    Article  MathSciNet  Google Scholar 

  49. Liu, Y.F., Hong, M., Dai, Y.H.: Max–min fairness linear transceiver design problem for a multi-user SIMO interference channel is polynomial time solvable. IEEE Signal Process. Lett. 20, 27–30 (2013)

    Article  Google Scholar 

  50. Liu, Y.F., Dai, Y.H., Luo, Z.Q.: Max–min fairness linear transceiver design for a multi-user MIMO interference channel. IEEE Trans. Signal Process. 61, 1413–1423 (2013)

    MathSciNet  Google Scholar 

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Acknowledgments

The authors wish to thank Professor Ya-Xiang Yuan and Professor Yu-Hong Dai of State Key Laboratory of Scientific and Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, for their helpful comments on the paper. The authors also thank Professor Guang-Lu Zhou of Department of Mathematics and Statistics, Curtin University, for sharing the code of Algorithm 1 in [1].

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Authors and Affiliations

  1. State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Ya-Feng Liu & Rui Diao

  2. Department of Applied Mathematics, Xidian University, Xi’an, 710071, China

    Feng Ye & Hong-Wei Liu

Authors
  1. Ya-Feng Liu
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  2. Rui Diao
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  3. Feng Ye
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  4. Hong-Wei Liu
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Corresponding author

Correspondence to Ya-Feng Liu.

Additional information

This work was partially supported by the National Natural Science Foundation of China (Nos. 11331012 and 11301516).

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Liu, YF., Diao, R., Ye, F. et al. An Efficient Inexact Newton-CG Algorithm for the Smallest Enclosing Ball Problem of Large Dimensions. J. Oper. Res. Soc. China 4, 167–191 (2016). https://doi.org/10.1007/s40305-015-0097-8

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