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Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VII. Inverse semigroup theory, closures, decomposition of perturbations

Robert Hildebrand (rhil***at***vt.edu)
Matthias Köppe (mkoeppe***at***math.ucdavis.edu)
Yuan Zhou (yuan.zhou***at***uky.edu)

Abstract: 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.

Keywords: cutting planes, group relaxation, #cutgeneratingfunctionology

Category 1: Integer Programming (Cutting Plane Approaches )


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Entry Submitted: 11/15/2018
Entry Accepted: 11/16/2018
Entry Last Modified: 01/05/2020

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