Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse Semigroup Theory, Closures, Decomposition of Perturbations
Open Journal of Mathematical Optimization, Volume 3 (2022), article no. 5, 44 p.

In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory–Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The infinite-dimensional subspaces have partial symmetries; to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper provides the foundation for grid-free algorithms for the Gomory–Johnson model, in particular for testing extremality of piecewise linear functions whose breakpoints are rational numbers with huge denominators.

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DOI: 10.5802/ojmo.16
Robert Hildebrand 1; Matthias Köppe 2; Yuan Zhou 3

1 Grado Dept. of Industrial and Systems Engineering, Virginia Tech
2 Dept. of Mathematics, University of California, Davis
3 Dept. of Mathematics, University of Kentucky
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Robert Hildebrand; Matthias Köppe; Yuan Zhou. Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse Semigroup Theory, Closures, Decomposition of Perturbations. Open Journal of Mathematical Optimization, Volume 3 (2022), article  no. 5, 44 p. doi : 10.5802/ojmo.16. https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.16/

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