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

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  • 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
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    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. 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.
    4. 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.
    5. 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.
    6. 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.
    7. B. Feijoo & R. R. Meyer, 1988. "Piecewise-Linear Approximation Methods for Nonseparable Convex Optimization," Management Science, INFORMS, vol. 34(3), pages 411-419, March.
    8. 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.
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    1. Ploussard, Quentin, 2024. "Piecewise linear approximation with minimum number of linear segments and minimum error: A fast approach to tighten and warm start the hierarchical mixed integer formulation," European Journal of Operational Research, Elsevier, vol. 315(1), pages 50-62.

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