IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v87y2023i1d10.1007_s10898-022-01267-4.html
   My bibliography  Save this article

Mixed-integer programming techniques for the minimum sum-of-squares clustering problem

Author

Listed:
  • Jan Pablo Burgard

    (Trier University)

  • Carina Moreira Costa

    (Trier University)

  • Christopher Hojny

    (Eindhoven University of Technology)

  • Thomas Kleinert

    (Quantagonia GmbH)

  • Martin Schmidt

    (Trier University)

Abstract

The minimum sum-of-squares clustering problem is a very important problem in data mining and machine learning with very many applications in, e.g., medicine or social sciences. However, it is known to be NP-hard in all relevant cases and to be notoriously hard to be solved to global optimality in practice. In this paper, we develop and test different tailored mixed-integer programming techniques to improve the performance of state-of-the-art MINLP solvers when applied to the problem—among them are cutting planes, propagation techniques, branching rules, or primal heuristics. Our extensive numerical study shows that our techniques significantly improve the performance of the open-source MINLP solver SCIP. Consequently, using our novel techniques, we can solve many instances that are not solvable with SCIP without our techniques and we obtain much smaller gaps for those instances that can still not be solved to global optimality.

Suggested Citation

  • Jan Pablo Burgard & Carina Moreira Costa & Christopher Hojny & Thomas Kleinert & Martin Schmidt, 2023. "Mixed-integer programming techniques for the minimum sum-of-squares clustering problem," Journal of Global Optimization, Springer, vol. 87(1), pages 133-189, September.
    • Handle: RePEc:spr:jglopt:v:87:y:2023:i:1:d:10.1007_s10898-022-01267-4
    DOI: 10.1007/s10898-022-01267-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-022-01267-4
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10898-022-01267-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Aihua Zheng & Bo Jiang & Yan Li & Xuehan Zhang & Chris Ding, 2017. "Elastic K-means using posterior probability," PLOS ONE, Public Library of Science, vol. 12(12), pages 1-16, December.
    2. Gerhard Reinelt, 1991. "TSPLIB—A Traveling Salesman Problem Library," INFORMS Journal on Computing, INFORMS, vol. 3(4), pages 376-384, November.
    3. Cuesta-Albertos, Juan Antonio & Fraiman, Ricardo, 2007. "Impartial trimmed k-means for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4864-4877, June.
    4. Seungwoo Han, 2022. "Spatial stratification and socio-spatial inequalities: the case of Seoul and Busan in South Korea," Palgrave Communications, Palgrave Macmillan, vol. 9(1), pages 1-14, December.
    5. Michael Brusco, 2006. "A Repetitive Branch-and-Bound Procedure for Minimum Within-Cluster Sums of Squares Partitioning," Psychometrika, Springer;The Psychometric Society, vol. 71(2), pages 347-363, June.
    6. Plastria, Frank, 2002. "Formulating logical implications in combinatorial optimisation," European Journal of Operational Research, Elsevier, vol. 140(2), pages 338-353, July.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. S Salhi & A Al-Khedhairi, 2010. "Integrating heuristic information into exact methods: The case of the vertex p-centre problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(11), pages 1619-1631, November.
    2. Rafael Blanquero & Emilio Carrizosa & Amaya Nogales-Gómez & Frank Plastria, 2014. "Single-facility huff location problems on networks," Annals of Operations Research, Springer, vol. 222(1), pages 175-195, November.
    3. Martins, Francisco Leonardo Bezerra & do Nascimento, José Cláudio, 2022. "Power law dynamics in genealogical graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 596(C).
    4. Marjan Marzban & Qian-Ping Gu & Xiaohua Jia, 2016. "New analysis and computational study for the planar connected dominating set problem," Journal of Combinatorial Optimization, Springer, vol. 32(1), pages 198-225, July.
    5. Ferrer, José M. & Martín-Campo, F. Javier & Ortuño, M. Teresa & Pedraza-Martínez, Alfonso J. & Tirado, Gregorio & Vitoriano, Begoña, 2018. "Multi-criteria optimization for last mile distribution of disaster relief aid: Test cases and applications," European Journal of Operational Research, Elsevier, vol. 269(2), pages 501-515.
    6. R. Baldacci & E. Hadjiconstantinou & A. Mingozzi, 2004. "An Exact Algorithm for the Capacitated Vehicle Routing Problem Based on a Two-Commodity Network Flow Formulation," Operations Research, INFORMS, vol. 52(5), pages 723-738, October.
    7. Roberto Tadei & Guido Perboli & Francesca Perfetti, 2017. "The multi-path Traveling Salesman Problem with stochastic travel costs," EURO Journal on Transportation and Logistics, Springer;EURO - The Association of European Operational Research Societies, vol. 6(1), pages 3-23, March.
    8. Lancia, Giuseppe & Vidoni, Paolo, 2020. "Finding the largest triangle in a graph in expected quadratic time," European Journal of Operational Research, Elsevier, vol. 286(2), pages 458-467.
    9. Oya Ekin Karaşan & A. Ridha Mahjoub & Onur Özkök & Hande Yaman, 2014. "Survivability in Hierarchical Telecommunications Networks Under Dual Homing," INFORMS Journal on Computing, INFORMS, vol. 26(1), pages 1-15, February.
    10. Paredes-Belmar, Germán & Montero, Elizabeth & Lüer-Villagra, Armin & Marianov, Vladimir & Araya-Sassi, Claudio, 2022. "Vehicle routing for milk collection with gradual blending: A case arising in Chile," European Journal of Operational Research, Elsevier, vol. 303(3), pages 1403-1416.
    11. Thanh Tan Doan & Nathalie Bostel & Minh Hoàng Hà & Vu Hoang Vuong Nguyen, 2023. "New mixed integer linear programming models and an iterated local search for the clustered traveling salesman problem with relaxed priority rule," Journal of Combinatorial Optimization, Springer, vol. 46(1), pages 1-27, August.
    12. Pop, Petrică C. & Cosma, Ovidiu & Sabo, Cosmin & Sitar, Corina Pop, 2024. "A comprehensive survey on the generalized traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 314(3), pages 819-835.
    13. F. Angel-Bello & Y. Cardona-Valdés & A. Álvarez, 2019. "Mixed integer formulations for the multiple minimum latency problem," Operational Research, Springer, vol. 19(2), pages 369-398, June.
    14. Jean-Charles Créput & Amir Hajjam & Abderrafiaa Koukam & Olivier Kuhn, 2012. "Self-organizing maps in population based metaheuristic to the dynamic vehicle routing problem," Journal of Combinatorial Optimization, Springer, vol. 24(4), pages 437-458, November.
    15. Jian Lin & Xiangfei Zeng & Jianxun Liu & Keqin Li, 2022. "Angular bisector insertion algorithm for solving small-scale symmetric and asymmetric traveling salesman problem," Journal of Combinatorial Optimization, Springer, vol. 43(1), pages 235-252, January.
    16. Leticia Vargas & Nicolas Jozefowiez & Sandra Ulrich Ngueveu, 2017. "A dynamic programming operator for tour location problems applied to the covering tour problem," Journal of Heuristics, Springer, vol. 23(1), pages 53-80, February.
    17. Afsaneh Amiri & Majid Salari, 2019. "Time-constrained maximal covering routing problem," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 41(2), pages 415-468, June.
    18. Marcel Turkensteen & Dmitry Malyshev & Boris Goldengorin & Panos M. Pardalos, 2017. "The reduction of computation times of upper and lower tolerances for selected combinatorial optimization problems," Journal of Global Optimization, Springer, vol. 68(3), pages 601-622, July.
    19. A. S. Santos & A. M. Madureira & M. L. R. Varela, 2018. "The Influence of Problem Specific Neighborhood Structures in Metaheuristics Performance," Journal of Mathematics, Hindawi, vol. 2018, pages 1-14, July.
    20. Markus Sinnl, 2021. "Mixed-integer programming approaches for the time-constrained maximal covering routing problem," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 43(2), pages 497-542, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jglopt:v:87:y:2023:i:1:d:10.1007_s10898-022-01267-4. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.