Open Journal of Mathematical Optimization

Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse Semigroup Theory, Closures, Decomposition of Perturbations 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. Digital Object Identifier An extended abstract of 13 pages titled On perturbation spaces of minimal valid functions: Inverse semigroup theory and equivariant decomposition theorem Integer Programming and Combinatorial Lecture preliminary of of the The authors DMS-2012764 (M. Köppe), DMS-2012429 (Y. Zhou). Part of this work was done while R. Hildebrand and M. Köppe were visiting the Simons Institute for the Theory of Computing in Fall 2017. It was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF grant CCF-1740425.