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.

Revised:

Accepted:

Published online:

^{1}; Matthias Köppe

^{2}; Yuan Zhou

^{3}

@article{OJMO_2022__3__A5_0, author = {Robert Hildebrand and Matthias K\"oppe and Yuan Zhou}, title = {Equivariant {Perturbation} in {Gomory} and {Johnson{\textquoteright}s} {Infinite} {Group} {Problem.} {VII.} {Inverse} {Semigroup} {Theory,} {Closures,} {Decomposition} of {Perturbations}}, journal = {Open Journal of Mathematical Optimization}, eid = {5}, publisher = {Universit\'e de Montpellier}, volume = {3}, year = {2022}, doi = {10.5802/ojmo.16}, language = {en}, url = {https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.16/} }

TY - JOUR TI - Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse Semigroup Theory, Closures, Decomposition of Perturbations JO - Open Journal of Mathematical Optimization PY - 2022 DA - 2022/// VL - 3 PB - Université de Montpellier UR - https://ojmo.centre-mersenne.org/articles/10.5802/ojmo.16/ UR - https://doi.org/10.5802/ojmo.16 DO - 10.5802/ojmo.16 LA - en ID - OJMO_2022__3__A5_0 ER -

%0 Journal Article %T Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse Semigroup Theory, Closures, Decomposition of Perturbations %J Open Journal of Mathematical Optimization %D 2022 %V 3 %I Université de Montpellier %U https://doi.org/10.5802/ojmo.16 %R 10.5802/ojmo.16 %G en %F OJMO_2022__3__A5_0

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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