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Sparsity of Lift-and-Project Cutting Planes

  • Matthias Walter
Conference paper
Part of the Operations Research Proceedings book series (ORP)

Abstract

It is well-known that sparsity (i.e. having only a few nonzero coefficients) is a desirable property for cutting planes in mixed-integer programming. We show that on the MIPLIB 2003 problem instance set, using only 10 very dense cutting planes (compared to thousands of constraints in a model), leads to a run time increase of 25 % on average for the LP-solver. We introduce the concept of dual sparsity (a property of the row-multipliers of the cut) and show a strong correlation between dual and primal (the usual) sparsity. Lift-and-project cuts crucially depend on the choice of a so-called normalization, of which we compared several known ones with respect to their actual and possible sparsity. Then a new normalization is tested that improves the dual (and hence the primal) sparsity of the generated cuts.

Keywords

Cutting Plane Normalization Constraint Rational Polyhedron Converse Concept Original Linear Program 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

I owe many thanks to my internship advisor Laci Ladanyi in the CPLEX group at IBM as well as my university advisor Volker Kaibel. Additionally, I want to thank Andrea Lodi, Tobias Achterberg and Roland Wunderling for several stimulating discussions.

References

  1. 1.
    Achterberg, T.: SCIP: solving constraint integer programs. Math. Program. Comput. 1(1), 141 (2009)CrossRefGoogle Scholar
  2. 2.
    Achterberg, T., Koch, T., Martin, A.: MIPLIB 2003. Operations Res. Lett. 34, 361–372 (2006)CrossRefGoogle Scholar
  3. 3.
    Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295324 (1993)Google Scholar
  4. 4.
    Balas, E., Perregaard, M.: A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer gomory cuts for 0–1 programming. Math. Program. 94, 221–245 (2003)CrossRefGoogle Scholar
  5. 5.
    Fischetti, M., Lodi, A., Tramontani, A.: On the separation of disjunctive cuts. Math. Program. 128, 205–230 (June 2011)Google Scholar
  6. 6.
    Wolter, K.: Implementation of cutting plane separators for mixed integer programs. Masters thesis (2006)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Otto-von-Guericke Universität MagdeburgMagdeburgGermany

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