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Geometric dual formulation for first-derivative-based univariate cubic L 1 splines

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Abstract

With the objective of generating “shape-preserving” smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based \(\mathcal{C}^1\)-smooth univariate cubic L 1 splines. An L 1 spline minimizes the L 1 norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating the coefficients of an L 1 spline is a nonsmooth non-linear convex program. Via Fenchel’s conjugate transformation, the geometric dual program is a smooth convex program with a linear objective function and convex cubic constraints. The dual-to-primal transformation is accomplished by solving a linear program.

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Authors and Affiliations

  1. Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100080, China

    Y. B. Zhao

  2. Industrial Engineering and Operations Research, North Carolina State University, Raleigh, NC, 27695-7906, USA

    S.-C. Fang

  3. Departments of Mathematical Sciences and Industrial Engineering, Tsinghua University, Beijing, China

    S.-C. Fang

  4. Mathematics Division, Army Research Office, Army Research Laboratory, Research Triangle Park, NC, 27709-2211, USA

    J. E. Lavery

Authors
  1. Y. B. Zhao
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  2. S.-C. Fang
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  3. J. E. Lavery
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Correspondence to S.-C. Fang.

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Zhao, Y.B., Fang, SC. & Lavery, J.E. Geometric dual formulation for first-derivative-based univariate cubic L 1 splines. J Glob Optim 40, 589–621 (2008). https://doi.org/10.1007/s10898-006-9124-y

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  • DOI: https://doi.org/10.1007/s10898-006-9124-y

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