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

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.