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Chebyshev model arithmetic for factorable functions

Author

Listed:
  • Jai Rajyaguru

    (Imperial College London)

  • Mario E. Villanueva

    (Imperial College London)

  • Boris Houska

    (ShanghaiTech University)

  • Benoît Chachuat

    (Imperial College London)

Abstract

This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden.

Suggested Citation

  • Jai Rajyaguru & Mario E. Villanueva & Boris Houska & Benoît Chachuat, 2017. "Chebyshev model arithmetic for factorable functions," Journal of Global Optimization, Springer, vol. 68(2), pages 413-438, June.
    • Handle: RePEc:spr:jglopt:v:68:y:2017:i:2:d:10.1007_s10898-016-0474-9
    DOI: 10.1007/s10898-016-0474-9
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    References listed on IDEAS

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    1. Achim Wechsung & Spencer Schaber & Paul Barton, 2014. "The cluster problem revisited," Journal of Global Optimization, Springer, vol. 58(3), pages 429-438, March.
    2. Agustín Bompadre & Alexander Mitsos & Benoît Chachuat, 2013. "Convergence analysis of Taylor models and McCormick-Taylor models," Journal of Global Optimization, Springer, vol. 57(1), pages 75-114, September.
    3. Agustín Bompadre & Alexander Mitsos, 2012. "Convergence rate of McCormick relaxations," Journal of Global Optimization, Springer, vol. 52(1), pages 1-28, January.
    4. A. Tsoukalas & A. Mitsos, 2014. "Multivariate McCormick relaxations," Journal of Global Optimization, Springer, vol. 59(2), pages 633-662, July.
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