23 research outputs found
Software for cut-generating functions in the Gomory--Johnson model and beyond
We present software for investigations with cut generating functions in the
Gomory-Johnson model and extensions, implemented in the computer algebra system
SageMath.Comment: 8 pages, 3 figures; to appear in Proc. International Congress on
Mathematical Software 201
New computer-based search strategies for extreme functions of the Gomory--Johnson infinite group problem
We describe new computer-based search strategies for extreme functions for
the Gomory--Johnson infinite group problem. They lead to the discovery of new
extreme functions, whose existence settles several open questions.Comment: 54 pages, many figure
Light on the Infinite Group Relaxation
This is a survey on the infinite group problem, an infinite-dimensional
relaxation of integer linear optimization problems introduced by Ralph Gomory
and Ellis Johnson in their groundbreaking papers titled "Some continuous
functions related to corner polyhedra I, II" [Math. Programming 3 (1972),
23-85, 359-389]. The survey presents the infinite group problem in the modern
context of cut generating functions. It focuses on the recent developments,
such as algorithms for testing extremality and breakthroughs for the k-row
problem for general k >= 1 that extend previous work on the single-row and
two-row problems. The survey also includes some previously unpublished results;
among other things, it unveils piecewise linear extreme functions with more
than four different slopes. An interactive companion program, implemented in
the open-source computer algebra package Sage, provides an updated compendium
of known extreme functions.Comment: 45 page
Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. VII. Inverse semigroup theory, closures, decomposition of perturbations
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.Comment: 67 pages, 21 figures; v2: changes to sections 10.2-10.3, improved
figures; v3: additional figures and minor updates, add reference to IPCO
abstract. CC-BY-S
The structure of the infinite models in integer programming
The infinite models in integer programming can be described as the convex
hull of some points or as the intersection of halfspaces derived from valid
functions. In this paper we study the relationships between these two
descriptions. Our results have implications for corner polyhedra. One
consequence is that nonnegative, continuous valid functions suffice to describe
corner polyhedra (with or without rational data)