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Globally optimal univariate spline approximations

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  • Robert Mohr

    (Karlsruhe Institute of Technology (KIT))

  • Maximilian Coblenz

    (Ludwigshafen University of Business and Society)

  • Peter Kirst

    (Wageningen University and Research (WUR))

Abstract

We revisit the problem of computing optimal spline approximations for univariate least-squares splines from a combinatorial optimization perspective. In contrast to most approaches from the literature we aim at globally optimal coefficients as well as a globally optimal placement of a fixed number of knots for a discrete variant of this problem. To achieve this, two different possibilities are developed. The first approach that we present is the formulation of the problem as a mixed-integer quadratically constrained problem, which can be solved using commercial optimization solvers. The second method that we propose is a branch-and-bound algorithm tailored specifically to the combinatorial formulation. We compare our algorithmic approaches empirically on both, real and synthetic curve fitting data sets from the literature. The numerical experiments show that our approach to tackle the least-squares spline approximation problem with free knots is able to compute solutions to problems of realistic sizes within reasonable computing times.

Suggested Citation

  • Robert Mohr & Maximilian Coblenz & Peter Kirst, 2023. "Globally optimal univariate spline approximations," Computational Optimization and Applications, Springer, vol. 85(2), pages 409-439, June.
    • Handle: RePEc:spr:coopap:v:85:y:2023:i:2:d:10.1007_s10589-023-00462-7
    DOI: 10.1007/s10589-023-00462-7
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    References listed on IDEAS

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    1. D. G. T. Denison & B. K. Mallick & A. F. M. Smith, 1998. "Automatic Bayesian curve fitting," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(2), pages 333-350.
    2. M. P. Wand, 2000. "A Comparison of Regression Spline Smoothing Procedures," Computational Statistics, Springer, vol. 15(4), pages 443-462, December.
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