37,451 research outputs found
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
Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. I. The One-Dimensional Case
We give an algorithm for testing the extremality of minimal valid functions
for Gomory and Johnson's infinite group problem that are piecewise linear
(possibly discontinuous) with rational breakpoints. This is the first set of
necessary and sufficient conditions that can be tested algorithmically for
deciding extremality in this important class of minimal valid functions. We
also present an extreme function that is a piecewise linear function with some
irrational breakpoints, whose extremality follows from a new principle.Comment: 38 pages, 10 figure
On the notions of facets, weak facets, and extreme functions of the Gomory-Johnson infinite group problem
We investigate three competing notions that generalize the notion of a facet
of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson
model. These notions were known to coincide for continuous piecewise linear
functions with rational breakpoints. We show that two of the notions, extreme
functions and facets, coincide for the case of continuous piecewise linear
functions, removing the hypothesis regarding rational breakpoints. We then
separate the three notions using discontinuous examples.Comment: 18 pages, 2 figure
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
On Minimal Valid Inequalities for Mixed Integer Conic Programs
We study disjunctive conic sets involving a general regular (closed, convex,
full dimensional, and pointed) cone K such as the nonnegative orthant, the
Lorentz cone or the positive semidefinite cone. In a unified framework, we
introduce K-minimal inequalities and show that under mild assumptions, these
inequalities together with the trivial cone-implied inequalities are sufficient
to describe the convex hull. We study the properties of K-minimal inequalities
by establishing algebraic necessary conditions for an inequality to be
K-minimal. This characterization leads to a broader algebraically defined class
of K- sublinear inequalities. We establish a close connection between
K-sublinear inequalities and the support functions of sets with a particular
structure. This connection results in practical ways of showing that a given
inequality is K-sublinear and K-minimal.
Our framework generalizes some of the results from the mixed integer linear
case. It is well known that the minimal inequalities for mixed integer linear
programs are generated by sublinear (positively homogeneous, subadditive and
convex) functions that are also piecewise linear. This result is easily
recovered by our analysis. Whenever possible we highlight the connections to
the existing literature. However, our study unveils that such a cut generating
function view treating the data associated with each individual variable
independently is not possible in the case of general cones other than
nonnegative orthant, even when the cone involved is the Lorentz cone
Structure and Interpretation of Dual-Feasible Functions
We study two techniques to obtain new families of classical and general
Dual-Feasible Functions: A conversion from minimal Gomory--Johnson functions;
and computer-based search using polyhedral computation and an automatic
maximality and extremality test.Comment: 6 pages extended abstract to appear in Proc. LAGOS 2017, with 21
pages of appendi
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
Cores of Cooperative Games in Information Theory
Cores of cooperative games are ubiquitous in information theory, and arise
most frequently in the characterization of fundamental limits in various
scenarios involving multiple users. Examples include classical settings in
network information theory such as Slepian-Wolf source coding and multiple
access channels, classical settings in statistics such as robust hypothesis
testing, and new settings at the intersection of networking and statistics such
as distributed estimation problems for sensor networks. Cooperative game theory
allows one to understand aspects of all of these problems from a fresh and
unifying perspective that treats users as players in a game, sometimes leading
to new insights. At the heart of these analyses are fundamental dualities that
have been long studied in the context of cooperative games; for information
theoretic purposes, these are dualities between information inequalities on the
one hand and properties of rate, capacity or other resource allocation regions
on the other.Comment: 12 pages, published at
http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/318704 in EURASIP
Journal on Wireless Communications and Networking, Special Issue on "Theory
and Applications in Multiuser/Multiterminal Communications", April 200
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