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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.

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Funding

Funding was provided by Solar Turbines.

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  1. University of California, San Diego, San Diego, USA

    Cody Allen & Mauricio de Oliveira

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  1. Cody Allen
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  2. Mauricio de Oliveira
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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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