Solving the 2-norm k-hyperplane clustering problem via multi-norm formulations

Stefano Coniglio

Keywords: hyperplane clustering; mathematical programming; spatial branch and bound

Abstract: We tackle the 2-norm (Euclidean) $k$-Hyperplane Clustering problem ($k$-HC$_2$), which asks for finding $k$ hyperplanes that minimize the sum of squared 2-norm (Euclidean) distances between each point and its closest hyperplane. We solve the problem to global optimality via spatial branch-and-bound techniques (SBB) by strengthening a mixed integer quadratically-constrained quadratic programming formulation with constraints that arise when formulating the problem in $p$-norms with $p \neq 2$. In particular, we show that, for every (appropriately scaled) $p \in \mathbb{N} \cup \{\infty\}$, one obtains a variant of $k$-HC$_2$, whose optimal solutions yield lower bounds within a multiplicative approximation factor. We focus on the case of polyhedral norms where $p=1, \infty$ (which admit a disjunctive-programming reformulation), and prove that strengthening the original formulation by including, on top of the original 2-norm constraints, the constraints of one of the polyhedral-norms leads to an SBB method where nonzero lower bounds are obtained in a linear (as opposed to exponential) number of SBB nodes. Experimentally, we show that our strengthened formulations lead to speedups from $\frac{1}{4}$ to 1.5 orders of magnitude, drastically improving the problem's solvability to global optimality.

Primary Area: optimization

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Submission Number: 13987