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SOS-SDP: An Exact Solver for Minimum Sum-of-Squares Clustering

Author

Listed:
  • Veronica Piccialli

    (University of Rome Tor Vergata, 00133 Roma RM, Italy)

  • Antonio M. Sudoso

    (University of Rome Tor Vergata, 00133 Roma RM, Italy)

  • Angelika Wiegele

    (Universität Klagenfurt, 9020 Klagenfurt, Austria)

Abstract

The minimum sum-of-squares clustering problem (MSSC) consists of partitioning n observations into k clusters in order to minimize the sum of squared distances from the points to the centroid of their cluster. In this paper, we propose an exact algorithm for the MSSC problem based on the branch-and-bound technique. The lower bound is computed by using a cutting-plane procedure in which valid inequalities are iteratively added to the Peng–Wei semidefinite programming (SDP) relaxation. The upper bound is computed with the constrained version of k -means in which the initial centroids are extracted from the solution of the SDP relaxation. In the branch-and-bound procedure, we incorporate instance-level must-link and cannot-link constraints to express knowledge about which data points should or should not be grouped together. We manage to reduce the size of the problem at each level, preserving the structure of the SDP problem itself. To the best of our knowledge, the obtained results show that the approach allows us to successfully solve, for the first time, real-world instances up to 4,000 data points.

Suggested Citation

  • Veronica Piccialli & Antonio M. Sudoso & Angelika Wiegele, 2022. "SOS-SDP: An Exact Solver for Minimum Sum-of-Squares Clustering," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2144-2162, July.
    • Handle: RePEc:inm:orijoc:v:34:y:2022:i:4:p:2144-2162
    DOI: 10.1287/ijoc.2022.1166
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    References listed on IDEAS

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    1. Frank de Meijer & Renata Sotirov, 2021. "SDP-Based Bounds for the Quadratic Cycle Cover Problem via Cutting-Plane Augmented Lagrangian Methods and Reinforcement Learning," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1262-1276, October.
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    3. Bissan Ghaddar & Miguel Anjos & Frauke Liers, 2011. "A branch-and-cut algorithm based on semidefinite programming for the minimum k-partition problem," Annals of Operations Research, Springer, vol. 188(1), pages 155-174, August.
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    6. Michael Brusco, 2006. "A Repetitive Branch-and-Bound Procedure for Minimum Within-Cluster Sums of Squares Partitioning," Psychometrika, Springer;The Psychometric Society, vol. 71(2), pages 347-363, June.
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