Skip to main content
SpringerLink
Log in

New heuristic for harmonic means clustering

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

It is well known that some local search heuristics for \(K\)-clustering problems, such as \(k\)-means heuristic for minimum sum-of-squares clustering occasionally stop at a solution with a smaller number of clusters than the desired number \(K\). Such solutions are called degenerate. In this paper, we reveal that the degeneracy also exists in \(K\)-harmonic means (KHM) method, proposed as an alternative to \(K\)-means heuristic, but which is less sensitive to the initial solution. In addition, we discover two types of degenerate solutions and provide examples for both. Based on these findings, we give a simple method to remove degeneracy during the execution of the KHM heuristic; it can be used as a part of any other heuristic for KHM clustering problem. We use KHM heuristic within a recent variant of variable neighborhood search (VNS) based heuristic. Extensive computational analysis, performed on test instances usually used in the literature, shows that significant improvements are obtained if our simple degeneracy correcting method is used within both KHM and VNS. Moreover, our VNS based heuristic suggested here may be considered as a new state-of-the-art heuristic for solving KHM clustering problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Alguwaizani, A., Hansen, P., Mladenovic, N., Ngai, E.: Variable neighborhood search for harmonic means clustering. Appl. Math. Model. 35, 2688–2694 (2011)

    Article  MATH  Google Scholar 

  2. Aloise, D., Deshpande, A., Hansen, P., Popat, P.: Np-hardness of Euclidean sum-of-squares clustering. Mach. Learn. 75, 245–248 (2009)

    Article  Google Scholar 

  3. Aloise, D., Hansen, P.: Clustering. In: Sheir D.r. (ed.) Handbook of Discrete and Combinatorial Mathemaics. CRC Press (2009)

  4. Bai, L., Liang, J., Dang, C., Cao, F.: A cluster centers initialization method for clustering categorical data. Expert Syst. Appl. 39, 8022–8029 (2012)

    Article  Google Scholar 

  5. Blake, C.L., Merz, C.J.: UCI repository of machine learning databases. http://archive.ics.uci.edu/ml/datasets.html (1998)

  6. Brimberg, J., Mladenović, N.: Degeneracy in the multi-source Weber problem. Math. Program. 85(1), 213–220 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brimberg, J., Hansen, P., Mladenovic, N.: Attraction probabilities in Variable neighborhood search. 4OR 8, 181–194 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cao, F., Liang, J., Jiang, G.: An initialization method for the \(K\)- Means algorithm using neighborhood model. Comput. Math. Appl. 58, 474–483 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Carrizosa, E., Mladenovic, N., Todosijevic, R.: Sum-of-squares clustering on networks. Yugosl. J. Oper. Res. 21, 157–161 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chua, Y.-M., Xiab, W.-F.: Two optimal double inequalities between power mean and logarithmic mean. Comput. Math. Appl. 60, 83–89 (2010)

    Article  MathSciNet  Google Scholar 

  11. Čižmešija, A.: A new sharp double inequality for generalized Heronian, harmonic and power means. Comput. Math. Appl. 64, 664671 (2012)

    Google Scholar 

  12. Erisoglu, M., Calis, N., Sakallioglu, S.: A new algorithm for initial cluster centers in \(k\)-means algorithm. Pattern Recognit. Lett. 32, 1701–1705 (2011)

    Article  Google Scholar 

  13. Hamerly, G., Elkan, C.: Alternatives to the \(k\)-means algorithm that find better clusterings. In: Proceedings of the Eleventh International Conference on Information and Knowledge Management, pp. 600–607. ACM (2002)

  14. Hansen, P., Jaumard, B., Mladenovic, N.: Minimum sum of squares clustering in a low dimensional space. J. Classif. 15, 37–56 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hansen, P., Mladenovic, N.: J-means: a new local search heuristic for minimum sum of squares clustering. Pattern Recognit. 34, 405–413 (2001)

    Article  MATH  Google Scholar 

  16. Hansen, P., Mladenovic, N., Pérez, J.A.M.: Variable neighbourhood search: methods and applications. 4-OR 6, 319–360 (2008)

    MATH  Google Scholar 

  17. Hua, J., Yi, S., Li, J., et al.: Ant clustering algorithm with K-harmonic means clustering. Expert Syst. Appl. 37, 8679–8684 (2010). doi:10.1016/j.eswa.2010.06.061

    Article  Google Scholar 

  18. Li, Q., Mitianoudis, N., Stathaki, T.: Spatial kernel K-harmonic means clustering for multi-spectral image segmentation. Image Process. IET 1(2), 156–167 (2007)

    Article  Google Scholar 

  19. Mladenović, N., Brimberg, J.: A degeneracy property in continuous location-allocation problems. Les Cahiers du GERAD, G-96-37 (1996)

  20. Mladenović, N., Hansen, P.: Variable neighbourhood search. Comput. Oper. Res. 24, 1097–1100 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Mladenovic, N., Todosijevic, R., Urosevic, D.: An efficient general variable neighborhood search for large TSP problem with time windows. Yugosl. J. Oper. Res. 23, 19–31 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  22. Pakhira, M.K.: A modified k-means Algorithm to avoid empty clusters. Int. J. Recent Trends Eng. 1, 220–226 (2009)

    Google Scholar 

  23. Ruspini, E.H.: Numerical methods for fuzzy clustering. Inf. Sci. 2, 319–350 (1970)

    Article  MATH  Google Scholar 

  24. Steinley, D., Brusco, M.J.: Initializing k-means batch clustering: a critical evaluation of several techniques. J. Classif. 24, 99–121 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Xu, R., Wunsch, D.: Clustering. IEEE Press, New York (2009)

    Google Scholar 

  26. Yang, Fengqin, Sun, Tieli, Zhang, Changhai: An efficient hybrid data clustering method based on K-harmonic means and particle swarm optimization original research article. Expert Syst. Appl. 36, 9847–9852 (2009)

    Article  Google Scholar 

  27. Yin, M., Hu, Y., Yang, F., et al.: A novel hybrid K-harmonic means and gravitational search algorithm approach for clustering. Expert Syst. Appl. 38, 9319–9324 (2011). doi:10.1016/j.eswa.2011.01.018

    Article  Google Scholar 

  28. Zhang, B.: Generalized k-harmonic means—boosting in unsupervised learning. Technical Report, HPL-2000-137, Hewlett-Packard Laboratories (2000)

  29. Zhang, B., Hsu, M., Dayal, U.: K-harmonic means—a data clustering algorithm. Technical Report, HPL-1999-124, Hewlett-Packard Laboratories (1999)

Download references

Acknowledgments

The research of E. Carrizosa is partially supported by Grants MTM2009-14039 (Ministerio de Educación y Ciencia, Spain) and FQM329 (Junta de Andalucía, Spain). Part of this research was done while N. Mladenović was visiting the Instituto de Matemáticas de la Universidad de Sevilla (Grant SAB2009-0144, Ministerio de Educación y Ciencia, Spain).

Author information

Authors and Affiliations

  1. Faculdad de Matematicas, Universidad de Seville, Seville, Spain

    Emilio Carrizosa

  2. Math department, King Faisal University, AlAhsa , Kingdom of Saudi Arabia

    Abdulrahman Alguwaizani

  3. GERAD and École des Hautes Etudes Commerciales, 3000 ch. de la Cote-Sainte-Catherine, Montreal , H3T 2A7, Canada

    Pierre Hansen

  4. LAMIH, University of Valenciennes, Le Mont Houy, 59313 , Valenciennes CEDEX 9, France

    Nenad Mladenović

Authors
  1. Emilio Carrizosa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdulrahman Alguwaizani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pierre Hansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nenad Mladenović
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Abdulrahman Alguwaizani.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carrizosa, E., Alguwaizani, A., Hansen, P. et al. New heuristic for harmonic means clustering. J Glob Optim 63, 427–443 (2015). https://doi.org/10.1007/s10898-014-0175-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-014-0175-1

Keywords

Search

Navigation