Mathematical Programming

, Volume 163, Issue 1–2, pp 301–358 | Cite as

Equivariant perturbation in Gomory and Johnson’s infinite group problem—III: foundations for the k-dimensional case with applications to \(k=2\)

  • Amitabh Basu
  • Robert Hildebrand
  • Matthias Köppe
Full Length Paper Series A


We develop foundational tools for classifying the extreme valid functions for the k-dimensional infinite group problem. In particular, we present the general regular solution to Cauchy’s additive functional equation on restricted lower-dimensional convex domains. This provides a k-dimensional generalization of the so-called Interval Lemma, allowing us to deduce affine properties of the function from certain additivity relations. Next, we study the discrete geometry of additivity domains of piecewise linear functions, providing a framework for finite tests of minimality and extremality. We then give a theory of non-extremality certificates in the form of perturbation functions. We apply these tools in the context of minimal valid functions for the two-dimensional infinite group problem that are piecewise linear on a standard triangulation of the plane, under a regularity condition called diagonal constrainedness. We show that the extremality of a minimal valid function is equivalent to the extremality of its restriction to a certain finite two-dimensional group problem. This gives an algorithm for testing the extremality of a given minimal valid function.

Mathematics Subject Classification

90C10 90C57 39B52 39B62 


  1. 1.
    Aczél, J.: Lectures on Functional Equations and Their Applications. Academic Press, London (1966)MATHGoogle Scholar
  2. 2.
    Aliprantis, C., Border, K.: Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer, Berlin (2006)MATHGoogle Scholar
  3. 3.
    Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: Maximal lattice-free convex sets in linear subspaces. Math. Oper. Res. 35, 704–720 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Basu, A., Conforti, M., Cornuéjols, G., Zambelli, G.: A counterexample to a conjecture of Gomory and Johnson. Math. Program. Ser. A 133(1–2), 25–38 (2012). doi: 10.1007/s10107-010-0407-1 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Basu, A., Hildebrand, R., Köppe, M.: Equivariant perturbation in Gomory and Johnson’s infinite group problem. I. The one-dimensional case. Math. Oper. Res. 40(1), 105–129 (2014). doi: 10.1287/moor.2014.0660 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Basu, A., Hildebrand, R., Köppe, M.: Equivariant perturbation in Gomory and Johnson’s infinite group problem. IV. The general unimodular two-dimensional case, Manuscript (2016)Google Scholar
  7. 7.
    Basu, A., Hildebrand, R., Köppe, M., Molinaro, M.: A \((k+1)\)-slope theorem for the \(k\)-dimensional infinite group relaxation. SIAM J. Optim. 23(2), 1021–1040 (2013). doi: 10.1137/110848608 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Conforti, M., Cornuéjols, G., Zambelli, G.: Corner polyhedra and intersection cuts. Surv. Oper. Res. Manag. Sci. 16, 105–120 (2011)MATHGoogle Scholar
  9. 9.
    Cornuéjols, G., Molinaro, M.: A 3-slope theorem for the infinite relaxation in the plane. Math. Program. 142(1–2), 83–105 (2013). doi: 10.1007/s10107-012-0562-7 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific, Singapore (2002)CrossRefMATHGoogle Scholar
  11. 11.
    Dey, S.S., Richard, J.-P.P.: Facets of two-dimensional infinite group problems. Math. Oper. Res. 33(1), 140–166 (2008). doi: 10.1287/moor.1070.0283 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dey, S.S., Richard, J.-P.P.: Relations between facets of low- and high-dimensional group problems. Math. Program. 123(2), 285–313 (2010). doi: 10.1007/s10107-009-0303-8 MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    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 (2010). doi: 10.1007/s10107-008-0229-6 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dhombres, J., Ger, R.: Conditional Cauchy equations. Glasnik Mat. 13(33), 39–62 (1978)MathSciNetMATHGoogle Scholar
  15. 15.
    Forster, W.: Homotopy methods. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 669–750. Kluwer Academic Publishers, Dordrecht (1995)CrossRefGoogle Scholar
  16. 16.
    Gomory, R.E.: Some polyhedra related to combinatorial problems. Linear Algebra Appl. 2, 451–558 (1969)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra, I. Math. Program. 3, 23–85 (1972). doi: 10.1007/BF01584976 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Gomory, R.E., Johnson, E.L.: Some continuous functions related to corner polyhedra, II. Math. Program. 3, 359–389 (1972). doi: 10.1007/BF01585008 MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gomory, R.E., Johnson, E.L.: T-space and cutting planes. Math. Program. 96, 341–375 (2003). doi: 10.1007/s10107-003-0389-3 MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hildebrand, R.: Algorithms and cutting planes for mixed integer programs. Ph.D. thesis, University of California, Davis (2013)Google Scholar
  21. 21.
    Kannappan, P.: Functional Equations and Inequalities with Applications. Springer, Berlin (2009)CrossRefMATHGoogle Scholar
  22. 22.
    Kuczma, M.: Functional equations on restricted domains. Aequationes Math. 18(1–2), 1–34 (1978). doi: 10.1007/BF01844065 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kuczma, M.: Introduction to the Theory of Functional Equations and Inequalities. Birkhäuser, Boston (2009)CrossRefMATHGoogle Scholar
  24. 24.
    Miller, L.A., Li, Y., Richard, J.-P.P.: New inequalities for finite and infinite group problems from approximate lifting. Naval Res. Logist. (NRL) 55(2), 172–191 (2008). doi: 10.1002/nav.20275 MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Radó, F., Baker, J.A.: Pexider’s equation and aggregation of allocations. Aequationes Math. 32(1), 227–239 (1987). doi: 10.1007/BF02311311 MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Richard, J.-P.P., Li, Y., Miller, L.A.: Valid inequalities for MIPs and group polyhedra from approximate liftings. Math. Program. 118(2), 253–277 (2009). doi: 10.1007/s10107-007-0190-9 MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  1. 1.Department of Applied Mathematics and StatisticsThe Johns Hopkins UniversityBaltimoreUSA
  2. 2.Mathematical SciencesIBM T.J. Watson Research CenterYorktown HeightsUSA
  3. 3.Department of MathematicsUniversity of California, DavisDavisUSA

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