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An Approximation Algorithm for Optimal Piecewise Linear Interpolations of Bounded Variable Products

Author

Listed:
  • Andreas Bärmann

    (Discrete Optimization)

  • Robert Burlacu

    (Discrete Optimization)

  • Lukas Hager

    (Discrete Optimization)

  • Katja Kutzer

    (Discrete Optimization)

Abstract

We investigate the optimal piecewise linear interpolation of the bivariate product xy over rectangular domains. More precisely, our aim is to minimize the number of simplices in the triangulation underlying the interpolation, while respecting a prescribed approximation error. First, we show how to construct optimal triangulations consisting of up to five simplices. Using these as building blocks, we construct a triangulation scheme called crossing swords that requires at most - times the number of simplices in any optimal triangulation. In other words, we derive an approximation algorithm for the optimal triangulation problem. We also show that crossing swords yields optimal triangulations in the case that each simplex has at least one axis-parallel edge. Furthermore, we present approximation guarantees for other well-known triangulation schemes, namely for the red refinement and longest-edge bisection strategies as well as for a generalized version of K1-triangulations. Thereby, we are able to show that our novel approach dominates previous triangulation schemes from the literature, which is underlined by illustrative numerical examples.

Suggested Citation

  • Andreas Bärmann & Robert Burlacu & Lukas Hager & Katja Kutzer, 2023. "An Approximation Algorithm for Optimal Piecewise Linear Interpolations of Bounded Variable Products," Journal of Optimization Theory and Applications, Springer, vol. 199(2), pages 569-599, November.
    • Handle: RePEc:spr:joptap:v:199:y:2023:i:2:d:10.1007_s10957-023-02292-3
    DOI: 10.1007/s10957-023-02292-3
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    References listed on IDEAS

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    1. Juan Pablo Vielma & Shabbir Ahmed & George Nemhauser, 2010. "Mixed-Integer Models for Nonseparable Piecewise-Linear Optimization: Unifying Framework and Extensions," Operations Research, INFORMS, vol. 58(2), pages 303-315, April.
    2. Steffen Rebennack & Josef Kallrath, 2015. "Continuous Piecewise Linear Delta-Approximations for Bivariate and Multivariate Functions," Journal of Optimization Theory and Applications, Springer, vol. 167(1), pages 102-117, October.
    3. Loay Alkhalifa & Hans Mittelmann, 2022. "New Algorithm to Solve Mixed Integer Quadratically Constrained Quadratic Programming Problems Using Piecewise Linear Approximation," Mathematics, MDPI, vol. 10(2), pages 1-15, January.
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