Skip to main content

A new efficient algorithm based on DC programming and DCA for clustering


In this paper, a version of K-median problem, one of the most popular and best studied clustering measures, is discussed. The model using squared Euclidean distances terms to which the K-means algorithm has been successfully applied is considered. A fast and robust algorithm based on DC (Difference of Convex functions) programming and DC Algorithms (DCA) is investigated. Preliminary numerical solutions on real-world databases show the efficiency and the superiority of the appropriate DCA with respect to the standard K-means algorithm.

This is a preview of subscription content, access via your institution.


  1. 1.

    Alon N., Spencer J.H. (1991): The Probabilistic Method. Wiley, New York, NY

    Google Scholar 

  2. 2.

    Arora, S., Kannan, R.: Learning mixtures of arbitrary Gaussians. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, July 6–8, pp. 247–257. Heraklion, Crete, Greece (2001)

  3. 3.

    Al-Sultan K. (1995): A Tabu search approach to the clustering problem. Pattern Recogn. 28(9): 443–1453

    Article  Google Scholar 

  4. 4.

    Bradley, P.S., Mangasarian, O.L., Street, W.N.: Clustering via concave minimization, Technical Report 96-03, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin May 1996. Advances In: Mozer, M.C., Jordan, M.I., Petsche, T. (eds.) Neural Information processing Systems 9, pp. 368–374. MIT Press, Cambridge, MA Available by

  5. 5.

    Bradley, B.S., Mangasarian, O.L.: Feature selection via concave minimization and support vector machines. In: Shavlik, J. (eds.) Machine Learning Proceedings of the Fifteenth International Conferences(ICML’98), pp. 82–90. San Francisco, CA 1998, Morgan Kaufmann.

  6. 6.

    Charikar, M., Guha, S.: Improved combinatorial algorithms for facility location and k-median problems. In: Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, 17–18 October, pp. 378–388. New York, NY, USA (1999)

  7. 7.

    Charikar, M., Guha, S., Tardos, E., Shmoys, D.B.: A constant-factor approximation algorithm for the k-median problem. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing pp. 1–10 (1999)

  8. 8.

    De Leeuw J. (1997): Applications of convex analysis to multidimensional scaling, Recent developments. In: Barra J.R., et al. (eds). Statistics. North-Holland Publishing company, Amsterdam, pp. 133–145

    Google Scholar 

  9. 9.

    De Leeuw J. (1988): Convergence of the majorization method for multidimensional scaling. J. Classi. 5, 163–180

    Article  Google Scholar 

  10. 10.

    Dhilon I.S., Korgan J., Nicholas C. (2003): Feature Selection and Document Clustering. In: Berry M.W. (eds). A Comprehensive Survey of Text Mining. Springer-Verlag, Berlin, pp. 73–100

    Google Scholar 

  11. 11.

    Duda R.O., Hart P.E. (1972): Pattern Classification and Scene Analysis. Wiley, New York

    Google Scholar 

  12. 12.

    Feder, T., Greene, D.: Optimal algorithms for approximate clustering. In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC) May 2–4, Chicago, Illinois, USA (1988)

  13. 13.

    Fisher D. (1988): Knowledge acquisition via incremental conceptual clusterin. Mach. Learning, 2, 139–172

    Google Scholar 

  14. 14.

    Fukunaga K. (1990): Statistical Pattern Recognition. Academic Press, NY

    Google Scholar 

  15. 15.

    Hiriart Urruty J.B., Lemarechal C. (1993): Convex Analysis and Minimization Algorithms. Springer Verlag, Berlin, Heidelberg

    Google Scholar 

  16. 16.

    Jain A.K., Dubes R.C. (1988): Algorithms for Clustering Data. Prentice-Hall Inc, Englewood Cliffs, NJ

    Google Scholar 

  17. 17.

    Hartigan J.A. (1975): Clustering Algorithms. Wiley, New York

    Google Scholar 

  18. 18.

    Le Thi Hoai An: Contribution à l’optimisation non convexe et l’optimisation globale: Théorie, Algoritmes et Applications. Habilitation à Diriger des Recherches, Université de Rouen, Juin (1997)

  19. 19.

    Le Thi Hoai An, Pham Dinh Tao (1997): Solving a class of linearly constrained indefinite quadratic problems by DC algorithms. J. Global Optim. 11(3): 253–285

    Article  Google Scholar 

  20. 20.

    Le Thi Hoai An, Pham Dinh Tao: DC programming approach for large-scale molecular optimization via the general distance geometry problem. Nonconvex Optimization and Its Applications, Special Issue “Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches”, pp. 301–339. Kluwer Academic Publishers, Dordrecht (2000) (This special issue contains refereed invited papers submitted at the conference on optimization in computational chemistry and molecular biology: local and global approaches held at Princeton University, 7–9 May 1999)

  21. 21.

    Le Thi Hoai An, Pham Dinh Tao (2001): DC programming approach for solving the multidimensional scaling problem. Nonconvex optimizations and its applications: special issue “From local to global optimization”, pp. 231–276. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  22. 22.

    Le Thi Hoai An, Pham Dinh Tao: DC programming: theory, algorithms and applications. The state of the art. In Proceedings of The First International Workshop on Global Constrained Optimization and Constraint Satisfaction (Cocos’ 02), 28 p. Valbonne-Sophia Antipolis, France, 2–4 October (2002)

  23. 23.

    Le Thi Hoai An, Pham Dinh Tao (2003): Large Scale Molecular Optimization from distances matrices by a DC optimization approach. SIAM J. Optim. 14(1): 77–116

    Article  Google Scholar 

  24. 24.

    Le Thi Hoai An, Pham Dinh Tao (2005): The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133, 23–46

    Article  Google Scholar 

  25. 25.

    Liu, Y., Shen, X., Doss, H.: Multicategory ψ–learning and support vector machine (29 p.). Conference on Machine Learning, Statistics and Discovery, June 22–26, Department of Statistics, the Ohio State University (2003)

  26. 26.

    Mangasarian O.L. (1997): Mathematical programming in data mining. Data Mining and Knowl. discov. 1, 183–201

    Article  Google Scholar 

  27. 27.

    MacQueen J.B. (1967): Some Methods for classification and analysis of multivariate observations. In: Proceedings of 5th Berkeley Symposium on Mathematical Statistics and Probability”, Berkeley, University of California Press,1, 281–297

  28. 28.

    Meyerson A., O’Callaghan L., Plotkin S. (2004): A k-Median algorithm with running time independent of data size. Machine Learn. 56, 61–87

    Article  Google Scholar 

  29. 29.

    Neumann, J., Schnörr, C., Steidl, G.: SVM-based feature selection by direct objective minimisation, Pattern Recognition. In: Proceeding of 26th DAGM Symposium, vol. 3175, pp. 212–219, LNCS (2004)

  30. 30.

    Pham Dinh Tao. (1984): Convergence of subgradient method for computing the bound-norm of matrices, Linear Algebra Appl. 62, 163–182

    Article  Google Scholar 

  31. 31.

    Pham Dinh Tao. (1984): Algorithmes de calcul d’une forme quadratique sur la boule unité de la norme du maximum. Numer. Math. 45, 377–440

    Article  Google Scholar 

  32. 32.

    Pham Dinh Tao, Le Thi Hoai An.(1997): Convex analysis approach to d.c. programming: Theory, Algorithms and Applications. Acta Mathematica Vietnamica, (dedicated to Professor Hoang Tuy on the occasion of his 70th birthday), 22(1): 289–355

    Google Scholar 

  33. 33.

    Pham Dinh Tao, Le Thi Hoai An. (1998): DC optimization algorithms for solving the trust region subproblem. SIAM J. Optim. 8, 476–505

    Article  Google Scholar 

  34. 34.

    Rao M.R. (1971): Cluster analysis and mathematical programming. J. Amer. Stat. Associ., 66, 622–626

    Article  Google Scholar 

  35. 35.

    Rockafellar R.T. (1970): Convex Analysis. Princeton University, Princeton

    Google Scholar 

  36. 36.

    Selim S.Z., Ismail M.A. (1984): K-means-Type algorithms: a generalized convergence theorem and characterization of local optimilaty. (Book Series) IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6, 81–87

    Article  Google Scholar 

  37. 37.

    Schüle T., Schnörr C., Weber S., Hornegger J. (2005): Discrete tomography by convex-concave regularization and d.c. programming. Discr. Appl. Math. 151, 229–243

    Article  Google Scholar 

  38. 38.

    Weber S., Schüle T., Schnörr C. (2005): Prior learning and convex-concave regularization of binary tomography. Electr. Notes in Discr. Math. 20, 313–327

    Article  Google Scholar 

  39. 39.

    Weber S., Schnörr C., Schüle T., Hornegger J., (2005): Binary Tomography by Iterating Linear Programs, Klette, R., Kozera, R., Noakes, L., and Weickert, J., (eds.) Computational Imaging and Vision – Geometric Properties from Incomplete Data, Kluwer Academic Press, Dordrecht

  40. 40.

    Wong, T., Katz, R., McCanne, S.: A preference clustering protocol for large-Scale Multicast Applications, Proceedings of the First International COST264 Workshop on Networked Group Communication, November 17–20, LNCS, pp. 1–18. Pisa, Italy (1999)

  41. 41.

    Wolberg W.H., Street W.N., Mangasarian O.L. (1995): Image analysis and machine learning applied to breast cancer diagnosis and prognosis. Anal. Quant. Cytol. Histol. 17(2): 77–87

    Google Scholar 

  42. 42.

    Wolberg W.H., Street W.N., Heisey D.M., Mangasarian O.L. (1995):Computerized breast cancer diagnosis and prognosis from fine-needle aspirates. Arch. Surg. 130, 511–516

    Google Scholar 

  43. 43.

    Wolberg W.H., Street W.N., Heisey D.M., Mangasarian O.L. (1995): Computer-derived nuclear features distinguish malignant from benign breast cytology. Hum. Patholo., 26, 792–796

    Article  Google Scholar 

  44. 44.

    Yuille A.L., Rangarajan A. (2002): The Concave-Convex Procedure (CCCP). Avances in Neural Information Processing Systems 14, pp. 1033–1040. MIT Press, Cambridge, MA

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Le Thi Hoai An.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

An, L.T.H., Belghiti, M.T. & Tao, P.D. A new efficient algorithm based on DC programming and DCA for clustering. J Glob Optim 37, 593–608 (2007).

Download citation


  • Clustering
  • K-median problem
  • K-means algorithm
  • DC programming
  • DCA
  • Nonsmooth nonconvex programming

AMS subject classifications

  • 65K05
  • 65K10
  • 90C26
  • 90C90
  • 15A60
  • 90C06