A distance-based point-reassignment heuristic for the k-hyperplane clustering problem

We consider the k-Hyperplane Clustering problem where, given a set of m   points in RnRn, we have to partition the set into k subsets (clusters) and determine a hyperplane for each of them, so as to minimize the sum of the squares of the Euclidean distances between the points and the hyperplane of the corresponding clusters. We give a nonconvex mixed-integer quadratically constrained quadratic programming formulation for the problem. Since even very small-size instances are challenging for state-of-the-art spatial branch-and-bound solvers like Couenne, we propose a heuristic in which many “critical” points are reassigned at each iteration. Such points, which are likely to be ill-assigned in the current solution, are identified using a distance-based criterion and their number is progressively decreased to zero. Our algorithm outperforms the best available one proposed by Bradley and Mangasarian on a set of real-world and structured randomly generated instances. For the largest instances, we obtain an average improvement in the solution quality of 54%.

► We study the k-Hyperplane Clustering problem (k-HC). ► The points are clustered w.r.t. hyperplanes rather than points or centers. ► We give a mixed-integer nonlinear programming formulation which we solve with Couenne. ► We propose the distance-based point reassignment heuristic (DBPR). ► Substantial improvements w.r.t. best available algorithm (up to 54% in solution quality).