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

Author

Listed:
  • Leo Liberti

    (CNRS LIX Ecole Polytechnique)

  • Benedetto Manca

    (Università degli Studi di Cagliari)

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.

Suggested Citation

  • Leo Liberti & Benedetto Manca, 2022. "Side-constrained minimum sum-of-squares clustering: mathematical programming and random projections," Journal of Global Optimization, Springer, vol. 83(1), pages 83-118, May.
    • Handle: RePEc:spr:jglopt:v:83:y:2022:i:1:d:10.1007_s10898-021-01047-6
    DOI: 10.1007/s10898-021-01047-6
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    References listed on IDEAS

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    1. Carl Eckart & Gale Young, 1936. "The approximation of one matrix by another of lower rank," Psychometrika, Springer;The Psychometric Society, vol. 1(3), pages 211-218, September.
    2. Leo Liberti & Fabrizio Marinelli, 2014. "Mathematical programming: Turing completeness and applications to software analysis," Journal of Combinatorial Optimization, Springer, vol. 28(1), pages 82-104, July.
    3. Ky Vu & Pierre-Louis Poirion & Leo Liberti, 2018. "Random Projections for Linear Programming," Mathematics of Operations Research, INFORMS, vol. 43(4), pages 1051-1071, November.
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