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A feasible rounding approach for mixed-integer optimization problems

Author

Listed:
  • Christoph Neumann

    (Karlsruhe Institute of Technology (KIT))

  • Oliver Stein

    (Karlsruhe Institute of Technology (KIT))

  • Nathan Sudermann-Merx

    (BASF Business Services GmbH)

Abstract

We introduce granularity as a sufficient condition for the consistency of a mixed-integer optimization problem, and show how to exploit it for the computation of feasible points: For optimization problems which are granular, solving certain linear problems and rounding their optimal points always leads to feasible points of the original mixed-integer problem. Thus, the resulting feasible rounding approach is deterministic and even efficient, i.e., it computes feasible points in polynomial time. The optimization problems appearing in the feasible rounding approaches have a structure that is similar to that of the continuous relaxation, and thus our approach has significant advantages over heuristics, as long as the problem is granular. For instance, the computational cost of our approach always corresponds to merely a single step of the feasibility pump. A computational study on optimization problems from the MIPLIB libraries demonstrates that granularity may be expected in various real world applications. Moreover, a comparison with Gurobi indicates that state of the art software does not always exploit granularity. Hence, our algorithms do not only possess a worst-case complexity advantage, but can also improve the CPU time needed to solve problems from practice.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:coopap:v:72:y:2019:i:2:d:10.1007_s10589-018-0042-y
    DOI: 10.1007/s10589-018-0042-y
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    References listed on IDEAS

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    1. Frederick S. Hillier, 1969. "Efficient Heuristic Procedures for Integer Linear Programming with an Interior," Operations Research, INFORMS, vol. 17(4), pages 600-637, August.
    2. 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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    Cited by:

    1. Charly Robinson La Rocca & Jean-François Cordeau & Emma Frejinger, 2024. "One-Shot Learning for MIPs with SOS1 Constraints," SN Operations Research Forum, Springer, vol. 5(3), pages 1-28, September.
    2. 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.

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