A Minimal Cardinality Solution to Fitting Sawtooth Piecewise-Linear Functions

• Published in: Journal of optimization theory and applications Vol. 192; no. 3; pp. 930 - 959
• Main Authors: Allen, Cody, de Oliveira, Mauricio
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-03-2022; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Combinatorial analysis; Efficiency; Engineering; Least squares method; Linear functions; Mathematics; Mathematics and Statistics; Noise; Operations Research/Decision Theory; Optimization; Parameters; Signal processing; Solvents; Theory of Computation; Piecewise-linear functions; 90C27; 62J07; 46n10; Least squares; Cardinality constraint
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-021-01998-6

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.