IDEAS home Printed from https://ideas.repec.org/a/inm/orijoc/v34y2022i2p1042-1047.html
   My bibliography  Save this article

A Comparison of Two Mixed-Integer Linear Programs for Piecewise Linear Function Fitting

Author

Listed:
  • John Alasdair Warwicker

    (Stochastic Optimization, Institute of Operations Research, Karlsruhe Institute of Technology, 76185 Karlsruhe, Baden-Württemberg, Germany)

  • Steffen Rebennack

    (Stochastic Optimization, Institute of Operations Research, Karlsruhe Institute of Technology, 76185 Karlsruhe, Baden-Württemberg, Germany)

Abstract

The problem of fitting continuous piecewise linear (PWL) functions to discrete data has applications in pattern recognition and engineering, amongst many other fields. To find an optimal PWL function, the positioning of the breakpoints connecting adjacent linear segments must not be constrained and should be allowed to be placed freely. Although the univariate PWL fitting problem has often been approached from a global optimisation perspective, recently, two mixed-integer linear programming approaches have been presented that solve for optimal PWL functions. In this paper, we compare the two approaches: the first was presented by Rebennack and Krasko [Rebennack S, Krasko V (2020) Piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. 32(2):507–530] and the second by Kong and Maravelias [Kong L, Maravelias CT (2020) On the derivation of continuous piecewise linear approximating functions. INFORMS J. Comput. 32(3):531–546]. Both formulations are similar in that they use binary variables and logical implications modelled by big- M constructs to ensure the continuity of the PWL function, yet the former model uses fewer binary variables. We present experimental results comparing the time taken to find optimal PWL functions with differing numbers of breakpoints across 10 data sets for three different objective functions. Although neither of the two formulations is superior on all data sets, the presented computational results suggest that the formulation presented by Rebennack and Krasko is faster. This might be explained by the fact that it contains fewer complicating binary variables and sparser constraints. Summary of Contribution: This paper presents a comparison of the mixed-integer linear programming models presented in two recent studies published in the INFORMS Journal on Computing . Because of the similarity of the formulations of the two models, it is not clear which one is preferable. We present a detailed comparison of the two formulations, including a series of comparative experimental results across 10 data sets that appeared across both papers. We hope that our results will allow readers to take an objective view as to which implementation they should use.

Suggested Citation

  • John Alasdair Warwicker & Steffen Rebennack, 2022. "A Comparison of Two Mixed-Integer Linear Programs for Piecewise Linear Function Fitting," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 1042-1047, March.
    • Handle: RePEc:inm:orijoc:v:34:y:2022:i:2:p:1042-1047
    DOI: 10.1287/ijoc.2021.1114
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/ijoc.2021.1114
    Download Restriction: no
    File URL: https://libkey.io/10.1287/ijoc.2021.1114?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
    ---><---

    References listed on IDEAS

    as
    1. Toriello, Alejandro & Vielma, Juan Pablo, 2012. "Fitting piecewise linear continuous functions," European Journal of Operational Research, Elsevier, vol. 219(1), pages 86-95.
    2. Steffen Rebennack & Vitaliy Krasko, 2020. "Piecewise Linear Function Fitting via Mixed-Integer Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 507-530, April.
    3. Lingxun Kong & Christos T. Maravelias, 2020. "On the Derivation of Continuous Piecewise Linear Approximating Functions," INFORMS Journal on Computing, INFORMS, vol. 32(3), pages 531-546, July.
    4. B. Feijoo & R. R. Meyer, 1988. "Piecewise-Linear Approximation Methods for Nonseparable Convex Optimization," Management Science, INFORMS, vol. 34(3), pages 411-419, March.
    5. Steffen Rebennack & Josef Kallrath, 2015. "Continuous Piecewise Linear Delta-Approximations for Bivariate and Multivariate Functions," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 102-117, October.
    6. Noam Goldberg & Youngdae Kim & Sven Leyffer & Thomas Veselka, 2014. "Adaptively refined dynamic program for linear spline regression," Computational Optimization and Applications, Springer, vol. 58(3), pages 523-541, July.
    7. Steffen Rebennack, 2016. "Computing tight bounds via piecewise linear functions through the example of circle cutting problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(1), pages 3-57, August.
    8. Matthias Walter, 2014. "Sparsity of Lift-and-Project Cutting Planes," Operations Research Proceedings, in: Stefan Helber & Michael Breitner & Daniel Rösch & Cornelia Schön & Johann-Matthias Graf von der Schu (ed.), Operations Research Proceedings 2012, edition 127, pages 9-14, Springer.
    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. Kazda, Kody & Li, Xiang, 2024. "A linear programming approach to difference-of-convex piecewise linear approximation," European Journal of Operational Research, Elsevier, vol. 312(2), pages 493-511.
    2. Steffen Rebennack & Vitaliy Krasko, 2020. "Piecewise Linear Function Fitting via Mixed-Integer Linear Programming," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 507-530, April.
    3. Steffen Rebennack, 2022. "Data-driven stochastic optimization for distributional ambiguity with integrated confidence region," Journal of Global Optimization, Springer, vol. 84(2), pages 255-293, October.
    4. Noam Goldberg & Steffen Rebennack & Youngdae Kim & Vitaliy Krasko & Sven Leyffer, 2021. "MINLP formulations for continuous piecewise linear function fitting," Computational Optimization and Applications, Springer, vol. 79(1), pages 223-233, May.
    5. David Lucas dos Santos Abreu & Erlon Cristian Finardi, 2022. "Continuous Piecewise Linear Approximation of Plant-Based Hydro Production Function for Generation Scheduling Problems," Energies, MDPI, vol. 15(5), pages 1-23, February.
    6. Nathan Sudermann-Merx & Steffen Rebennack, 2021. "Leveraged least trimmed absolute deviations," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 43(3), pages 809-834, September.
    7. Cody Allen & Mauricio Oliveira, 2022. "A Minimal Cardinality Solution to Fitting Sawtooth Piecewise-Linear Functions," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 930-959, March.
    8. Aloïs Duguet & Christian Artigues & Laurent Houssin & Sandra Ulrich Ngueveu, 2022. "Properties, Extensions and Application of Piecewise Linearization for Euclidean Norm Optimization in $$\mathbb {R}^2$$ R 2," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 418-448, November.
    9. Aakil M. Caunhye & Douglas Alem, 2023. "Practicable robust stochastic optimization under divergence measures with an application to equitable humanitarian response planning," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 45(3), pages 759-806, September.
    10. Mohammadi Fathabad, Abolhassan & Cheng, Jianqiang & Pan, Kai & Yang, Boshi, 2023. "Asymptotically tight conic approximations for chance-constrained AC optimal power flow," European Journal of Operational Research, Elsevier, vol. 305(2), pages 738-753.
    11. Jon Lee & Daphne Skipper & Emily Speakman & Luze Xu, 2023. "Gaining or Losing Perspective for Piecewise-Linear Under-Estimators of Convex Univariate Functions," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 1-35, January.
    12. Gambella, Claudio & Ghaddar, Bissan & Naoum-Sawaya, Joe, 2021. "Optimization problems for machine learning: A survey," European Journal of Operational Research, Elsevier, vol. 290(3), pages 807-828.
    13. Andreas Bärmann & Robert Burlacu & Lukas Hager & Thomas Kleinert, 2023. "On piecewise linear approximations of bilinear terms: structural comparison of univariate and bivariate mixed-integer programming formulations," Journal of Global Optimization, Springer, vol. 85(4), pages 789-819, April.
    14. Liu, Zhiyuan & Chen, Xinyuan & Hu, Jintao & Wang, Shuaian & Zhang, Kai & Zhang, Honggang, 2023. "An alternating direction method of multipliers for solving user equilibrium problem," European Journal of Operational Research, Elsevier, vol. 310(3), pages 1072-1084.
    15. Lambert, Mathieu & Hassani, Rachid, 2023. "Diesel genset optimization in remote microgrids," Applied Energy, Elsevier, vol. 340(C).
    16. Huaiyu Zhu & Yun Pan & Kwang-Ting Cheng & Ruohong Huan, 2018. "A lightweight piecewise linear synthesis method for standard 12-lead ECG signals based on adaptive region segmentation," PLOS ONE, Public Library of Science, vol. 13(10), pages 1-22, October.
    17. F. J. Hwang & Yao-Huei Huang, 2021. "An effective logarithmic formulation for piecewise linearization requiring no inequality constraint," Computational Optimization and Applications, Springer, vol. 79(3), pages 601-631, July.
    18. Liu, Zhiyuan & Zhang, Honggang & Zhang, Kai & Zhou, Zihan, 2023. "Integrating alternating direction method of multipliers and bush for solving the traffic assignment problem," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 177(C).
    19. Jan Szczegielniak & Krzysztof J Latawiec & Jacek Łuniewski & Rafał Stanisławski & Katarzyna Bogacz & Marcin Krajczy & Marek Rydel, 2018. "A study on nonlinear estimation of submaximal effort tolerance based on the generalized MET concept and the 6MWT in pulmonary rehabilitation," PLOS ONE, Public Library of Science, vol. 13(2), pages 1-18, February.
    20. Maximilian Roth & Georg Franke & Stephan Rinderknecht, 2022. "A Comprehensive Approach for an Approximative Integration of Nonlinear-Bivariate Functions in Mixed-Integer Linear Programming Models," Mathematics, MDPI, vol. 10(13), pages 1-17, 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:inm:orijoc:v:34:y:2022:i:2:p:1042-1047. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.