Column Generation for the Minimum Hyperplanes Clustering Problem

Given n points in ℝ^d and a maximum allowed tolerance ϵ > 0, the minimum hyperplanes clustering problem consists in finding a minimum number of hyperplanes such that the Euclidean distance between each point and the nearest hyperplane is at most ϵ. We present a column generation approach for this problem based on a mixed integer nonlinear formulation in which the master is a set covering problem and the pricing subproblem is a mixed integer program with a nonconvex normalization constraint. We propose different ways of generating the initial pool of columns and investigate their impact on the overall algorithm. Since the pricing subproblem is substantially complicated by the ℓ₂-norm constraint, we consider approximate pricing subproblems involving different norms. Some strategies for refining the solution and speeding-up the overall method are also discussed. The performance of our column generation algorithm is assessed on realistic randomly generated instances as well as on real-world instances.