Advertisement

The Structure of the Infinite Models in Integer Programming

  • Amitabh Basu
  • Michele Conforti
  • Marco Di SummaEmail author
  • Joseph Paat
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10328)

Abstract

The infinite models in integer programming can be described as the convex hull of some points or as the intersection of halfspaces derived from valid functions. In this paper we study the relationships between these two descriptions. Our results have implications for finite dimensional corner polyhedra. One consequence is that nonnegative continuous functions suffice to describe finite dimensional corner polyhedra with rational data. We also discover new facts about corner polyhedra with non-rational data.

Keywords

Valid Inequality Rational Vector Valid Function Strict Subset Recession Cone 
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.

References

  1. 1.
    Aczél, J., Dhombres, J.G.: Functional Equations in Several Variables. Encyclopedia of Mathematics and Its Applications, vol. 31. Cambridge University Press (1989)Google Scholar
  2. 2.
    Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35, 704–720 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Basu, A., Hildebrand, R., Köppe, M.: Light on the infinite group relaxation I: foundations and taxonomy. 4OR 14(1), 1–40 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Basu, A., Hildebrand, R., Köppe, M.: Light on the infinite group relaxation II: sufficient conditions for extremality, sequences, and algorithms. 4OR 14(2), 1–25 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Basu, A., Paat, J.: Operations that preserve the covering property of the lifting region. SIAM J. Optim. 25(4), 2313–2333 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Conforti, M., Cornuéjols, G., Zambelli, G.: Integer Programming, vol. 271. Springer, Switzerland (2014)zbMATHGoogle Scholar
  7. 7.
    Dash, S., Günlük, O.: Valid inequalities based on simple mixed-integer sets. Math. Program. 105, 29–53 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dey, S.S., Richard, J.P.P., Li, Y., Miller, L.A.: On the extreme inequalities of infinite group problems. Math. Program. 121(1), 145–170 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dickson, L.E.: Finiteness of the odd perfect and primitive abundant numbers with \(n\) distinct prime factors. Am. J. Math. 35(4), 413–422 (1913)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gomory, R.E.: Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2(4), 451–558 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra, I. Math. Program. 3, 23–85 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra, II. Math. Program. 3, 359–389 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hildebrand, R.: Algorithms and cutting planes for mixed integer programs. Ph.D. thesis, University of California, Davis, June 2013Google Scholar
  14. 14.
    Johnson, E.L.: On the group problem for mixed integer programming. Math. Program. Study 2, 137–179 (1974)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Köppe, M., Zhou, Y.: An electronic compendium of extreme functions for the gomory-johnson infinite group problem. Oper. Res. Lett. 43(4), 438–444 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Grundlehren der mathematischen Wissenschaften, vol. 305 (1996)Google Scholar
  17. 17.
    Letchford, A.N., Lodi, A.: Strengthening Chvátal-Gomory cuts and gomory fractional cuts. Oper. Res. Lett. 30(2), 74–82 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Miller, L.A., Li, Y., Richard, J.P.P.: New inequalities for finite and infinite group problems from approximate lifting. Naval Res. Logistics (NRL) 55(2), 172–191 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yıldız, S., Cornuéjols, G.: Cut-generating functions for integer variables. Math. Oper. Res. 41, 1381–1403 (2016)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Amitabh Basu
    • 1
  • Michele Conforti
    • 2
  • Marco Di Summa
    • 2
    Email author
  • Joseph Paat
    • 1
  1. 1.Department of Applied Mathematics and StatisticsThe Johns Hopkins UniversityBaltimoreUSA
  2. 2.Dipartimento di MatematicaUniversità degli Studi di PadovaPaduaItaly

Personalised recommendations