A Method for Best $$L_1$$ L 1 Data Approximation That Achieves Convexity–Concavity

Ioannis C. Demetriou ()
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Ioannis C. Demetriou: National and Kapodistrian University of Athens

Chapter 5 in Convex and Variational Analysis with Applications, 2026, pp 99-110 from Springer

Abstract: Abstract Let n measurements from a univariate process be given, which suggest that a potential shape of the underlying relation is convex–concave, but the data have lost the convexity–concavity property due to errors. We address the problem of making the least sum of moduli change to the measurements so that the second divided differences of the smoothed values change sign once. Hence the piecewise linear interpolant to the fit is composed of one convex and one concave section. Since the position of the sign change is also an unknown of this problem, the optimization calculation is nonlinear. It is proved that the required fit consists of two separate sections. One section whose second divided differences are nonnegative and one section whose second divided differences are nonpositive. Therefore, the required fit may be obtained by solving a linear programming problem on each section. Then a method is proposed that calculates the required fit by employing at most $$2n-4$$ 2 n - 4 linear programming calculations over subranges of the data.

Keywords: Convex–concave data smoothing; Least absolute deviations; Linear programming; Nonlinear data fitting; Piecewise linear approximation; Second divided differences (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-07860-5_5

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