A feasible rounding approach for mixed-integer optimization problems

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.

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Acknowledgements

The authors are grateful to Sven Leyffer, Benjamin Müller and two anonymous referees for their precise and substantial remarks, which helped to significantly improve the paper.

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Correspondence to Oliver Stein.

Appendix

Appendix

See Table 4

Table 4 A comparison of the feasible rounding approaches and Gurobi with regard to time (seconds) and objective value on unaltered models

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Neumann, C., Stein, O. & Sudermann-Merx, N. A feasible rounding approach for mixed-integer optimization problems. Comput Optim Appl 72, 309–337 (2019). https://doi.org/10.1007/s10589-018-0042-y

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Keywords

  • Rounding
  • Granularity
  • Inner parallel set
  • Consistency

Mathematics Subject Classification

  • 90C11
  • 90C10