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Mathematical programming formulations for piecewise polynomial functions

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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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    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. Achim Wechsung & Paul Barton, 2014. "Global optimization of bounded factorable functions with discontinuities," Journal of Global Optimization, Springer, vol. 58(1), pages 1-30, January.
    3. Nadia Martinez & Hadis Anahideh & Jay M. Rosenberger & Diana Martinez & Victoria C. P. Chen & Bo Ping Wang, 2017. "Global optimization of non-convex piecewise linear regression splines," Journal of Global Optimization, Springer, vol. 68(3), pages 563-586, July.
    4. Holmberg, Kaj, 1994. "Solving the staircase cost facility location problem with decomposition and piecewise linearization," European Journal of Operational Research, Elsevier, vol. 75(1), pages 41-61, May.
    5. Israel Zang, 1981. "Discontinuous Optimization by Smoothing," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 140-152, February.
    6. Stefan Scholtes, 2004. "Nonconvex Structures in Nonlinear Programming," Operations Research, INFORMS, vol. 52(3), pages 368-383, June.
    7. Beaumont, Nicholas, 1990. "An algorithm for disjunctive programs," European Journal of Operational Research, Elsevier, vol. 48(3), pages 362-371, October.
    8. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    9. Ahmet B. Keha & Ismael R. de Farias & George L. Nemhauser, 2006. "A Branch-and-Cut Algorithm Without Binary Variables for Nonconvex Piecewise Linear Optimization," Operations Research, INFORMS, vol. 54(5), pages 847-858, October.
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    Cited by:

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