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. 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.
    2. Gerhard Reinelt, 1991. "TSPLIB—A Traveling Salesman Problem Library," INFORMS Journal on Computing, INFORMS, vol. 3(4), pages 376-384, November.
    3. 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.
    4. 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.
    5. Plastria, Frank, 2002. "Formulating logical implications in combinatorial optimisation," European Journal of Operational Research, Elsevier, vol. 140(2), pages 338-353, July.
    6. 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.
    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. William Cook & Sanjeeb Dash & Ricardo Fukasawa & Marcos Goycoolea, 2009. "Numerically Safe Gomory Mixed-Integer Cuts," INFORMS Journal on Computing, INFORMS, vol. 21(4), pages 641-649, November.
    2. Thiago Serra & Ryan J. O’Neil, 2020. "MIPLIBing: Seamless Benchmarking of Mathematical Optimization Problems and Metadata Extensions," SN Operations Research Forum, Springer, vol. 1(3), pages 1-6, September.
    3. Barbato, Michele & Gouveia, Luís, 2024. "The Hamiltonian p-median problem: Polyhedral results and branch-and-cut algorithms," European Journal of Operational Research, Elsevier, vol. 316(2), pages 473-487.
    4. Saïd Echchakoui, 2020. "Why and how to merge Scopus and Web of Science during bibliometric analysis: the case of sales force literature from 1912 to 2019," Journal of Marketing Analytics, Palgrave Macmillan, vol. 8(3), pages 165-184, September.
    5. Balma, Ali & Salem, Safa Ben & Mrad, Mehdi & Ladhari, Talel, 2018. "Strong multi-commodity flow formulations for the asymmetric traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 271(1), pages 72-79.
    6. 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.
    7. Marilène Cherkesly & Claudio Contardo, 2021. "The conditional p-dispersion problem," Journal of Global Optimization, Springer, vol. 81(1), pages 23-83, September.
    8. Malaguti, Enrico & Martello, Silvano & Santini, Alberto, 2018. "The traveling salesman problem with pickups, deliveries, and draft limits," Omega, Elsevier, vol. 74(C), pages 50-58.
    9. Bernardino, Raquel & Paias, Ana, 2018. "Solving the family traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 267(2), pages 453-466.
    10. Ernst Althaus & Felix Rauterberg & Sarah Ziegler, 2020. "Computing Euclidean Steiner trees over segments," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 8(3), pages 309-325, October.
    11. Saïd Echchakoui, 0. "Why and how to merge Scopus and Web of Science during bibliometric analysis: the case of sales force literature from 1912 to 2019," Journal of Marketing Analytics, Palgrave Macmillan, vol. 0, pages 1-20.
    12. William Cook & Daniel G. Espinoza & Marcos Goycoolea, 2007. "Computing with Domino-Parity Inequalities for the Traveling Salesman Problem (TSP)," INFORMS Journal on Computing, INFORMS, vol. 19(3), pages 356-365, August.
    13. 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.
    14. Sattar Sattari & Mohammad Izadi, 2017. "An exact algorithm for the minimum dilation triangulation problem," Journal of Global Optimization, Springer, vol. 69(2), pages 343-367, October.
    15. Matteo Fischetti & Juan José Salazar González & Paolo Toth, 1998. "Solving the Orienteering Problem through Branch-and-Cut," INFORMS Journal on Computing, INFORMS, vol. 10(2), pages 133-148, May.
    16. 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).
    17. 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.
    18. Alejandro Toriello & William B. Haskell & Michael Poremba, 2014. "A Dynamic Traveling Salesman Problem with Stochastic Arc Costs," Operations Research, INFORMS, vol. 62(5), pages 1107-1125, October.
    19. 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.
    20. 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.

    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.