A two-phase heuristic for the bottleneck k-hyperplane clustering problem

Abstract

In the bottleneck hyperplane clustering problem, given n points in $\mathbb{R}^{d}$ and an integer k with 1≤k≤n, we wish to determine k hyperplanes and assign each point to a hyperplane so as to minimize the maximum Euclidean distance between each point and its assigned hyperplane. This mixed-integer nonlinear problem has several interesting applications but is computationally challenging due, among others, to the nonconvexity arising from the ℓ ₂-norm. After comparing several linear approximations to deal with the ℓ ₂-norm constraint, we propose a two-phase heuristic. First, an approximate solution is obtained by exploiting the ℓ _∞-approximation and the problem geometry, and then it is converted into an ℓ ₂-approximate solution. Computational experiments on realistic randomly generated instances and instances arising from piecewise affine maps show that our heuristic provides good quality solutions in a reasonable amount of time.