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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

Abstract

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 

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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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