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Lagrangian relaxation and pegging test for the clique partitioning problem

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  • Noriyoshi Sukegawa
  • Yoshitsugu Yamamoto
  • Liyuan Zhang

Abstract

The clique partitioning problem is an NP-hard combinatorial optimization problem with applications to data analysis such as clustering. Though a binary integer linear programming formulation has been known for years, one needs to deal with a huge number of variables and constraints when solving a large instance. In this paper, we propose a size reduction algorithm which is based on the Lagrangian relaxation and the pegging test, and verify its validity through numerical experiments. We modify the conventional subgradient method in order to manage the high dimensionality of the Lagrangian multipliers, and also make an improvement on the ordinary pegging test by taking advantage of the structural property of the clique partitioning problem. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Noriyoshi Sukegawa & Yoshitsugu Yamamoto & Liyuan Zhang, 2013. "Lagrangian relaxation and pegging test for the clique partitioning problem," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 7(4), pages 363-391, December.
    • Handle: RePEc:spr:advdac:v:7:y:2013:i:4:p:363-391
    DOI: 10.1007/s11634-013-0135-5
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    References listed on IDEAS

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    1. Rogers, David F. & Kulkarni, Shailesh S., 2005. "Optimal bivariate clustering and a genetic algorithm with an application in cellular manufacturing," European Journal of Operational Research, Elsevier, vol. 160(2), pages 423-444, January.
    2. Larsson, Torbjorn & Patriksson, Michael & Stromberg, Ann-Brith, 1996. "Conditional subgradient optimization -- Theory and applications," European Journal of Operational Research, Elsevier, vol. 88(2), pages 382-403, January.
    3. Gary Kochenberger & Fred Glover & Bahram Alidaee & Haibo Wang, 2005. "Clustering of Microarray data via Clique Partitioning," Journal of Combinatorial Optimization, Springer, vol. 10(1), pages 77-92, August.
    4. Saul Amorim & Jean-Pierre Barthélemy & Celso Ribeiro, 1992. "Clustering and clique partitioning: Simulated annealing and tabu search approaches," Journal of Classification, Springer;The Classification Society, vol. 9(1), pages 17-41, January.
    5. King, John R, 1980. "Machine-component group formation in group technology," Omega, Elsevier, vol. 8(2), pages 193-199.
    6. Egon Balas & Maria C. Carrera, 1996. "A Dynamic Subgradient-Based Branch-and-Bound Procedure for Set Covering," Operations Research, INFORMS, vol. 44(6), pages 875-890, December.
    7. Ravi Kumar, K. & Kusiak, Andrew & Vannelli, Anthony, 1986. "Grouping of parts and components in flexible manufacturing systems," European Journal of Operational Research, Elsevier, vol. 24(3), pages 387-397, March.
    8. Michael Brusco & Hans-Friedrich Köhn, 2009. "Clustering Qualitative Data Based on Binary Equivalence Relations: Neighborhood Search Heuristics for the Clique Partitioning Problem," Psychometrika, Springer;The Psychometric Society, vol. 74(4), pages 685-703, December.
    9. Marshall L. Fisher, 1981. "The Lagrangian Relaxation Method for Solving Integer Programming Problems," Management Science, INFORMS, vol. 27(1), pages 1-18, January.
    10. Ulrich Dorndorf & Erwin Pesch, 1994. "Fast Clustering Algorithms," INFORMS Journal on Computing, INFORMS, vol. 6(2), pages 141-153, May.
    11. Ulrich Dorndorf & Florian Jaehn & Erwin Pesch, 2008. "Modelling Robust Flight-Gate Scheduling as a Clique Partitioning Problem," Transportation Science, INFORMS, vol. 42(3), pages 292-301, August.
    12. Robert M. Nauss, 1976. "An Efficient Algorithm for the 0-1 Knapsack Problem," Management Science, INFORMS, vol. 23(1), pages 27-31, September.
    13. You, Byungjun & Yamada, Takeo, 2007. "A pegging approach to the precedence-constrained knapsack problem," European Journal of Operational Research, Elsevier, vol. 183(2), pages 618-632, December.
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    Cited by:

    1. Ramazan Ünlü & Petros Xanthopoulos, 2019. "A weighted framework for unsupervised ensemble learning based on internal quality measures," Annals of Operations Research, Springer, vol. 276(1), pages 229-247, May.

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