Advanced Search
MyIDEAS: Login to save this article or follow this journal

Cluster Analysis: An Application of Lagrangian Relaxation


Author Info

  • John M. Mulvey

    (Princeton University)

  • Harlan P. Crowder

    (IBM T. J. Watson Research Center)

Registered author(s):


    This paper presents and tests an effective optimization algorithm for clustering homogeneous data. The algorithm iteratively employs a subgradient method for determining lower bounds and a simple search procedure for determining upper bounds. The overall objective is to assign n objects to m mutually exclusive "clusters" such that the sum of the distances from each object to a designated cluster median is minimum. The model represents a special case of the uncapacitated facility location and m-median problems. This technique has proven efficient for examples with n \le 200 (i.e., the number of 0-1 variables \le 40,000); computational experiences with 10 real-world clustering applications are provided. A comparison with a hierarchical agglomerative heuristic, the minimum squared error method, is included. It is shown that the optimization algorithm is an effective solution technique for the homogeneous clustering problem, and also a good method for providing tight lower bounds for evaluating the quality of solutions generated by other procedures.

    Download Info

    If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
    File URL:
    Download Restriction: no

    Bibliographic Info

    Article provided by INFORMS in its journal Management Science.

    Volume (Year): 25 (1979)
    Issue (Month): 4 (April)
    Pages: 329-340

    as in new window
    Handle: RePEc:inm:ormnsc:v:25:y:1979:i:4:p:329-340

    Contact details of provider:
    Postal: 7240 Parkway Drive, Suite 300, Hanover, MD 21076 USA
    Phone: +1-443-757-3500
    Fax: 443-757-3515
    Web page:
    More information through EDIRC

    Related research

    Keywords: cluster analysis; integer programming applications; Lagrangian relaxation; heuristics;


    No references listed on IDEAS
    You can help add them by filling out this form.


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as in new window

    Cited by:
    1. Lau, Kin-nam & Leung, Pui-lam & Tse, Ka-kit, 1999. "A mathematical programming approach to clusterwise regression model and its extensions," European Journal of Operational Research, Elsevier, vol. 116(3), pages 640-652, August.
    2. Chen, Mu-Chen & Wu, Hsiao-Pin, 2005. "An association-based clustering approach to order batching considering customer demand patterns," Omega, Elsevier, vol. 33(4), pages 333-343, August.
    3. Mladenovic, Nenad & Brimberg, Jack & Hansen, Pierre & Moreno-Perez, Jose A., 2007. "The p-median problem: A survey of metaheuristic approaches," European Journal of Operational Research, Elsevier, vol. 179(3), pages 927-939, June.
    4. Maravalle, Maurizio & Simeone, Bruno & Naldini, Rosella, 1997. "Clustering on trees," Computational Statistics & Data Analysis, Elsevier, vol. 24(2), pages 217-234, April.
    5. Mangiameli, Paul & Chen, Shaw K. & West, David, 1996. "A comparison of SOM neural network and hierarchical clustering methods," European Journal of Operational Research, Elsevier, vol. 93(2), pages 402-417, September.
    6. Holmberg, Kaj, 1997. "Mean value cross decomposition applied to integer programming problems," European Journal of Operational Research, Elsevier, vol. 97(1), pages 124-138, February.
    7. Stefano Benati & Sergio García, 2012. "A p-median problem with distance selection," Statistics and Econometrics Working Papers ws121913, Universidad Carlos III, Departamento de Estadística y Econometría.
    8. Sáez-Aguado, Jesús & Trandafir, Paula Camelia, 2012. "Some heuristic methods for solving p-median problems with a coverage constraint," European Journal of Operational Research, Elsevier, vol. 220(2), pages 320-327.
    9. Michael Brusco & Hans-Friedrich Köhn, 2009. "Exemplar-Based Clustering via Simulated Annealing," Psychometrika, Springer, vol. 74(3), pages 457-475, September.
    10. Chen, Ja-Shen & Heragu, Sunderesh S., 1999. "Stepwise decomposition approaches for large scale cell formation problems," European Journal of Operational Research, Elsevier, vol. 113(1), pages 64-79, February.
    11. Wojtek Krzanowski & Glenn Milligan & Stanley Wasserman & Joseph Galaskiewicz & Joel Levine & Elke Weber & Peter Fishburn & Theodore Crovello & Bernard Baum & Wayne DeSarbo, 1987. "Book reviews," Journal of Classification, Springer, vol. 4(1), pages 111-141, March.
    12. Lozano, S. & Guerrero, F. & Onieva, L. & Larraneta, J., 1998. "Kohonen maps for solving a class of location-allocation problems," European Journal of Operational Research, Elsevier, vol. 108(1), pages 106-117, July.
    13. D'Alfonso, Thomas H. & Ventura, Jose A., 1995. "Assignment of tools to machines in a flexible manufacturing system," European Journal of Operational Research, Elsevier, vol. 81(1), pages 115-133, February.
    14. Wayne DeSarbo & Vijay Mahajan, 1984. "Constrained classification: The use of a priori information in cluster analysis," Psychometrika, Springer, vol. 49(2), pages 187-215, June.
    15. Michael Brusco & Douglas Steinley, 2011. "A Tabu-Search Heuristic for Deterministic Two-Mode Blockmodeling of Binary Network Matrices," Psychometrika, Springer, vol. 76(4), pages 612-633, October.
    16. Klose, Andreas & Drexl, Andreas, 2005. "Facility location models for distribution system design," European Journal of Operational Research, Elsevier, vol. 162(1), pages 4-29, April.


    This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.


    Access and download statistics


    When requesting a correction, please mention this item's handle: RePEc:inm:ormnsc:v:25:y:1979:i:4:p:329-340. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Mirko Janc).

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If references are entirely missing, you can add them using this form.

    If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.