Side-constrained minimum sum-of-squares clustering: mathematical programming and random projections

Leo Liberti

;Benedetto Manca

2021-01-01

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Abstract

This paper investigates a mathematical programming based methodology for solving the minimum sum-of-squares clustering problem, also known as the "k-means problem", in the presence of side constraints. We propose several exact and approximate mixed-integer linear and nonlinear formulations. The approximations are based on norm inequalities and random projections, the approximation guarantees of which are based on an additive version of the Johnson-Lindenstrauss lemma. We perform computational testing (with fixed CPU time) on a range of randomly generated and real data instances of medium size, but with high dimensionality. We show that when side constraints make k-means inapplicable, our proposed methodology-which is easy and fast to implement and deploy-can obtain good solutions in limited amounts of time.