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

TL;DR

This paper introduces SOS-SDP, an exact branch-and-bound algorithm for minimum sum-of-squares clustering that effectively solves real-world instances with up to 4000 data points by integrating SDP relaxations and constraints.

Contribution

The paper presents a novel exact solver combining SDP relaxations, cutting-plane methods, and constraint integration for large-scale MSSC problems.

Findings

Successfully solves real-world MSSC instances up to 4000 data points.

Integrates must-link and cannot-link constraints into the SDP-based branch-and-bound.

Reduces problem size at each branch level while preserving SDP structure.

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 where valid inequalities are iteratively added to the Peng-Wei SDP relaxation. The upper bound is computed with the constrained version of k-means where 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. The obtained results show that the approach allows to successfully solve for the first time real-world instances up to 4000 data points.