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Cluster Analysis: An Application of Lagrangian Relaxation

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Author Info

  • John M. Mulvey

    (Princeton University)

  • Harlan P. Crowder

    (IBM T. J. Watson Research Center)

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    Abstract

    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.

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    File URL: http://dx.doi.org/10.1287/mnsc.25.4.329
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    Bibliographic Info

    Article provided by INFORMS in its journal Management Science.

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

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    Handle: RePEc:inm:ormnsc:v:25:y:1979:i:4:p:329-340

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    Related research

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

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    Cited by:
    1. Michael Brusco & Hans-Friedrich Köhn, 2009. "Exemplar-Based Clustering via Simulated Annealing," Psychometrika, Springer, vol. 74(3), pages 457-475, September.
    2. 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.
    3. 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.
    4. Maravalle, Maurizio & Simeone, Bruno & Naldini, Rosella, 1997. "Clustering on trees," Computational Statistics & Data Analysis, Elsevier, vol. 24(2), pages 217-234, April.
    5. 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.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. 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.

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