IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v192y2022i3d10.1007_s10957-021-01998-6.html
   My bibliography  Save this article

A Minimal Cardinality Solution to Fitting Sawtooth Piecewise-Linear Functions

Author

Listed:
  • Cody Allen

    (University of California, San Diego)

  • Mauricio Oliveira

    (University of California, San Diego)

Abstract

In this paper, we explore a method to parameterize a linear function with jump discontinuities, which we refer to as a “sawtooth” function, and then develop theory and algorithms for estimating the function parameters from noisy data in a least-squares framework. Because there will always exist a sawtooth function that exactly fits a given data set, one is led to bounding the maximum number of jumps the sawtooth function can have in order to obtain reasonable practical estimates. The main contribution of the paper is a proof that cardinality of the optimal solutions to a relaxation of the associated least-squares problem in which a constraint on the cardinality of the solutions is replaced by a 1-norm constraint on the vector of jumps is a monotonic function of the parameter of the relaxation. This property allows one to avoid dealing with the combinatorial cardinality constraint and quickly explore solutions using the proposed convex relaxation. A fitting algorithm based on the proposed results is developed and illustrated with a simple numerical example.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:192:y:2022:i:3:d:10.1007_s10957-021-01998-6
    DOI: 10.1007/s10957-021-01998-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-021-01998-6
    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/s10957-021-01998-6?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. 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 & 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.
    3. Tsunokawa, Koji & Schofer, Joseph L., 1994. "Trend curve optimal control model for highway pavement maintenance: Case study and evaluation," Transportation Research Part A: Policy and Practice, Elsevier, vol. 28(2), pages 151-166, March.
    4. Tsoutsanis, Elias & Meskin, Nader & Benammar, Mohieddine & Khorasani, Khashayar, 2014. "A component map tuning method for performance prediction and diagnostics of gas turbine compressors," Applied Energy, Elsevier, vol. 135(C), pages 572-585.
    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. 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.
    4. Qiao, Julie Yu & Du, Runjia & Labi, Samuel & Fricker, Jon D. & Sinha, Kumares C., 2021. "Policy implications of standalone timing versus holistic timing of infrastructure interventions: Findings based on pavement surface roughness," Transportation Research Part A: Policy and Practice, Elsevier, vol. 148(C), pages 79-99.
    5. Kim, Jeong Ho & Kim, Tong Seop, 2019. "A new approach to generate turbine map data in the sub-idle operation regime of gas turbines," Energy, Elsevier, vol. 173(C), pages 772-784.
    6. 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.
    7. Lee, Jinwoo & Madanat, Samer, 2015. "A joint bottom-up solution methodology for system-level pavement rehabilitation and reconstruction," Transportation Research Part B: Methodological, Elsevier, vol. 78(C), pages 106-122.
    8. Seyedshohadaie, S. Reza & Damnjanovic, Ivan & Butenko, Sergiy, 2010. "Risk-based maintenance and rehabilitation decisions for transportation infrastructure networks," Transportation Research Part A: Policy and Practice, Elsevier, vol. 44(4), pages 236-248, May.
    9. 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.
    10. Son, Seongmin & Jeong, Yongju & Cho, Seong Kuk & Lee, Jeong Ik, 2020. "Development of supercritical CO2 turbomachinery off-design model using 1D mean-line method and Deep Neural Network," Applied Energy, Elsevier, vol. 263(C).
    11. 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.
    12. 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.
    13. Sathaye, Nakul & Madanat, Samer, 2011. "A bottom-up solution for the multi-facility optimal pavement resurfacing problem," Transportation Research Part B: Methodological, Elsevier, vol. 45(7), pages 1004-1017, August.
    14. Kang, Do Won & Kim, Tong Seop, 2018. "Model-based performance diagnostics of heavy-duty gas turbines using compressor map adaptation," Applied Energy, Elsevier, vol. 212(C), pages 1345-1359.
    15. Lee, Jinwoo & Madanat, Samer, 2014. "Joint optimization of pavement design, resurfacing and maintenance strategies with history-dependent deterioration models," Transportation Research Part B: Methodological, Elsevier, vol. 68(C), pages 141-153.
    16. Cheng, Xianda & Zheng, Haoran & Yang, Qian & Zheng, Peiying & Dong, Wei, 2023. "Surrogate model-based real-time gas path fault diagnosis for gas turbines under transient conditions," Energy, Elsevier, vol. 278(PA).
    17. 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.
    18. 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.
    19. Zhi-Chun Li & Dian Sheng, 2014. "Pavement rehabilitation scheduling and toll pricing under different regulatory regimes," Annals of Operations Research, Springer, vol. 217(1), pages 337-355, June.
    20. 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.

    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:joptap:v:192:y:2022:i:3:d:10.1007_s10957-021-01998-6. 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.