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A Minimal Cardinality Solution to Fitting Sawtooth Piecewise-Linear Functions

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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.

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Notes

  1. The proof follows trivially by induction and is omitted for brevity.

  2. The matrix used again in the numerical example in Sect. 4.

References

  1. Ahmadi, H., Martí, J.R., Moshref, A.: Piecewise linear approximation of generators cost functions using max-affine functions. In: 2013 IEEE Power & Energy Society General Meeting, pp. 1–5. IEEE (2013)

  2. Allen, C.W.: Remote Health Monitoring of Gas Turbines: Mathematical Modeling and Numerical Optimization Techniques. Ph.D. Dissertation, University of California, San Diego (2020)

  3. Allen, C.W., Holcomb, C.M., de Oliveira, M.: Estimating recoverable performance degradation rates and optimizing maintenance scheduling. J. Eng. Gas Turbines Power 141(1), 011032 (2019)

    Article  Google Scholar 

  4. Bemporad, A., Breschi, V., Piga, D., Boyd, S.: Fitting jump models. Automatica 96, 11–21 (2018)

    Article  MathSciNet  Google Scholar 

  5. Berkelaar, A.B., Roos, K., Terlaky, T., et al.: The optimal set and optimal partition approach to linear and quadratic programming. In: Gal, T., Greenberg, H.J. (eds.) Advances in Sensitivity Analysis and Parametric Programming, pp. 159–202. Springer, Boston, MA (1997)

  6. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press (2004)

  7. Flø, N.E., Faramarzi, L., de Cazenove, T., Hvidsten, O.A., et al.: Results from MEA degradation and reclaiming processes at the CO\(_2\) technology centre Mongstad. Energy Procedia 114, 1307–1324 (2017)

    Article  Google Scholar 

  8. Friedman, J., Hastie, T., Tibshirani, R.: The Elements of Statistical Learning, vol. 10. Springer Series in Statistics, New York (2001)

  9. Gill, P.E., Murray, W., Wright, M.H.: Practical Optimization. Society for Industrial and Applied Mathematics (2019)

  10. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press (2013)

  11. Ingle, A., Bucklew, J., Sethares, W., Varghese, T.: Slope estimation in noisy piecewise linear functions. Signal Process. 108, 576–588 (2015)

    Article  Google Scholar 

  12. Jordan, D.C., Kurtz, S.R., VanSant, K., Newmiller, J.: Compendium of photovoltaic degradation rates. Prog. Photovolt. Res. Appl. 24(7), 978–989 (2016)

    Article  Google Scholar 

  13. Löfberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei (2004)

  14. Logan, K.P.: Using a ship’s propeller for hull condition monitoring. Naval Eng. J. 124(1), 71–87 (2012)

    Google Scholar 

  15. Magnani, A., Boyd, S.: Convex piecewise-linear fitting. Optim. Eng. 10(1), 1–17 (2009)

    Article  MathSciNet  Google Scholar 

  16. Misener, R., Floudas, C.A.: Piecewise-linear approximations of multidimensional functions. J. Optim. Theory Appl. 145(1), 120–147 (2010)

    Article  MathSciNet  Google Scholar 

  17. Murphy, K.P.: Machine Learning: A Probabilistic Perspective. MIT Press (2012)

  18. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research. Springer (2006)

  19. Rebennack, S., Kallrath, J.: Continuous piecewise linear delta-approximations for bivariate and multivariate functions. J. Optim. Theory Appl. 167(1), 102–117 (2015)

    Article  MathSciNet  Google Scholar 

  20. Tibshirani, R.: Regression shrinkage and selection via the LASSO. J. R. Stat. Soc. Ser. B (Methodol.) 58(1), 267–288 (1996)

    MathSciNet  MATH  Google Scholar 

  21. Toriello, A., Vielma, J.P.: Fitting piecewise linear continuous functions. Eur. J. Oper. Res. 219(1), 86–95 (2012)

    Article  MathSciNet  Google Scholar 

  22. Tsoutsanis, E., Meskin, N., Benammar, M., Khorasani, K.: A component map tuning method for performance prediction and diagnostics of gas turbine compressors. Appl. Energy 135, 572–585 (2014)

    Article  Google Scholar 

  23. Tsunokawa, K., Schofer, J.L.: Trend curve optimal control model for highway pavement maintenance: case study and evaluation. Transp. Res. Part A Policy Pract. 28(2), 151–166 (1994)

    Article  Google Scholar 

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Funding was provided by Solar Turbines.

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Correspondence to Cody Allen.

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Communicated by Guoyin Li.

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Allen, C., de Oliveira, M. A Minimal Cardinality Solution to Fitting Sawtooth Piecewise-Linear Functions. J Optim Theory Appl 192, 930–959 (2022). https://doi.org/10.1007/s10957-021-01998-6

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