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Granularity in Nonlinear Mixed-Integer Optimization

Author

Listed:
  • Christoph Neumann

    (Karlsruhe Institute of Technology (KIT))

  • Oliver Stein

    (Karlsruhe Institute of Technology (KIT))

  • Nathan Sudermann-Merx

    (BASF SE)

Abstract

We study a new technique to check the existence of feasible points for mixed-integer nonlinear optimization problems that satisfy a structural requirement called granularity. For granular optimization problems, we show how rounding the optimal points of certain purely continuous optimization problems can lead to feasible points of the original mixed-integer nonlinear problem. To this end, we generalize results for the mixed-integer linear case from Neumann et al. (Comput Optim Appl 72:309–337, 2019). We study some additional issues caused by nonlinearity and show how to overcome them by extending the standard granularity concept to an advanced version, which we call pseudo-granularity. In a computational study on instances from a standard test library, we demonstrate that pseudo-granularity can be expected in many nonlinear applications from practice, and that its explicit use can be beneficial.

Suggested Citation

  • Christoph Neumann & Oliver Stein & Nathan Sudermann-Merx, 2020. "Granularity in Nonlinear Mixed-Integer Optimization," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 433-465, February.
    • Handle: RePEc:spr:joptap:v:184:y:2020:i:2:d:10.1007_s10957-019-01591-y
    DOI: 10.1007/s10957-019-01591-y
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    References listed on IDEAS

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    1. Christoph Neumann & Oliver Stein & Nathan Sudermann-Merx, 2019. "A feasible rounding approach for mixed-integer optimization problems," Computational Optimization and Applications, Springer, vol. 72(2), pages 309-337, March.
    2. C. Audet & P. Hansen & B. Jaumard & G. Savard, 1997. "Links Between Linear Bilevel and Mixed 0–1 Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 273-300, May.
    3. Michael R. Bussieck & Arne Stolbjerg Drud & Alexander Meeraus, 2003. "MINLPLib—A Collection of Test Models for Mixed-Integer Nonlinear Programming," INFORMS Journal on Computing, INFORMS, vol. 15(1), pages 114-119, February.
    4. Pierre Bonami & João Gonçalves, 2012. "Heuristics for convex mixed integer nonlinear programs," Computational Optimization and Applications, Springer, vol. 51(2), pages 729-747, March.
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