Sparsity of Lift-and-Project Cutting Planes

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


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.


Cutting Plane Normalization Constraint Rational Polyhedron Converse Concept Original Linear Program 
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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.


  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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