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Abstract

We study the uniqueness of minimal liftings of cut generating functions obtained from maximal lattice-free polytopes. We prove a basic invariance property of unique minimal liftings for general maximal lattice-free polytopes. This generalizes a previous result by Basu, Cornuéjols and Köppe [3] for simplicial maximal lattice-free polytopes, thus completely settling this fundamental question about lifting. We also extend results from [3] for minimal liftings in maximal lattice-free simplices to more general polytopes. These nontrivial generalizations require the use of deep theorems from discrete geometry and geometry of numbers, such as the Venkov-Alexandrov-McMullen theorem on translative tilings, and McMullen’s characterization of zonotopes.

Keywords

Integer Point Relative Interior Tile Space Unimodular Transformation Nontrivial Generalization 
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.

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References

  1. 1.
    Balas, E., Jeroslow, R.G.: Strengthening cuts for mixed integer programs. European Journal of Operational Research 4, 224–234 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Basu, A., Campelo, M., Conforti, M., Cornuejols, G., Zambelli, G.: Unique lifting of integer variables in minimal inequalities. Mathematical Programming A 141, 561–576 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Basu, A., Cornuéjols, G., Köppe, M.: Unique Minimal Liftings for Simplicial Polytopes. Mathematics of Operations Research 37(2), 346–355 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Conforti, M., Cornuéjols, G., Zambelli, G.: A Geometric Perspective on Lifting. Operations Research 59, 569–577 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Conforti, M., Cornuéjols, G., Zambelli, G.: Corner Polyhedron and Intersection Cuts. Surveys in Operations Research and Management Science 16, 105–120 (2011)CrossRefGoogle Scholar
  6. 6.
    Del Pia, A., Weismantel, R.: Relaxations of mixed integer sets from lattice-free polyhedra. 4OR 10(3), 221–244 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Dey, S.S., Wolsey, L.A.: Two Row Mixed Integer Cuts Via Lifting. Mathematical Programming B 124, 143–174 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Dey, S.S., Wolsey, L.A.: Composite Lifting of Group Inequalities and an Application to Two-Row Mixing Inequalities. Discrete Optimization 7, 256–268 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Lovász, L.: Geometry of Numbers and Integer Programming. In: Iri, M., Tanabe, K. (eds.) Mathematical Programming: Recent Developments and Applications, pp. 177–210. Kluwer (1989)Google Scholar
  10. 10.
    McMullen, P.: Polytopes with centrally symmetric faces. Israel Journal of Mathematics 8(2), 194–196 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    McMullen, P.: personal communicationGoogle Scholar
  12. 12.
    Gruber, P.M.: Convex and Discrete Geometry, Grundlehren der mathematischen Wissenschaften 336. Springer (2007)Google Scholar
  13. 13.
    Ziegler, G.: Lectures on Polytopes. Springer (1995)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Gennadiy Averkov
    • 1
  • Amitabh Basu
    • 2
  1. 1.Institute of Mathematical Optimization, Faculty of MathematicsUniversity of MagdeburgGermany
  2. 2.Department of Applied Mathematics and StatisticsThe Johns Hopkins UniversityUSA

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