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Mathematical programming formulations for piecewise polynomial functions

Author

Listed:
  • Bjarne Grimstad

    (NTNU
    Solution Seeker)

  • Brage R. Knudsen

    (NTNU
    SINTEF Energy Research)

Abstract

This paper studies mathematical programming formulations for solving optimization problems with piecewise polynomial (PWP) constraints. We elaborate on suitable polynomial bases as a means of efficiently representing PWPs in mathematical programs, comparing and drawing connections between the monomial basis, the Bernstein basis, and B-splines. The theory is presented for both continuous and semi-continuous PWPs. Using a disjunctive formulation, we then exploit the characteristic of common polynomial basis functions to significantly reduce the number of nonlinearities, and to suggest a bound-tightening technique for PWP constraints. We derive several extensions using Bernstein cuts, an expanded Bernstein basis, and an expanded monomial basis, which upon a standard big-M reformulation yield a set of new MINLP models. The formulations are compared by globally solving six test sets of MINLPs and a realistic petroleum production optimization problem. The proposed framework shows promising numerical performance and facilitates the solution of PWP-constrained optimization problems using standard MINLP software.

Suggested Citation

  • Bjarne Grimstad & Brage R. Knudsen, 2020. "Mathematical programming formulations for piecewise polynomial functions," Journal of Global Optimization, Springer, vol. 77(3), pages 455-486, July.
    • Handle: RePEc:spr:jglopt:v:77:y:2020:i:3:d:10.1007_s10898-020-00881-4
    DOI: 10.1007/s10898-020-00881-4
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    References listed on IDEAS

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    Cited by:

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