115 research outputs found
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
The Dynamics of Metropolitan Housing Prices
This article is the winner of the Innovative Thinking ââThinking Out of the Boxââ manuscript prize (sponsored by the Homer Hoyt Advanced Studies Institute) presented at the 2001 American Real Estate Society Annual Meeting. This study examines the dynamics of real housing price appreciation in 130 metropolitan areas across the United States. The study finds that real housing price appreciation is strongly influenced by the growth of population and real changes in income, construction costs and interest rates. The study also finds that stock market appreciation imparts a strong current and lagged wealth effect on housing prices. Housing appreciation rates also are found to vary across areas because of location-specific fixed-effects; these fixed effects represent the residuals of housing price appreciation attributable to location. The magnitudes of the fixed-effects in particular cities are positively correlated with restrictive growth management policies and limitations on land availability.
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)
Approximation of corner polyhedra with families of intersection cuts
We study the problem of approximating the corner polyhedron using
intersection cuts derived from families of lattice-free sets in .
In particular, we look at the problem of characterizing families that
approximate the corner polyhedron up to a constant factor, which depends only
on and not the data or dimension of the corner polyhedron. The literature
already contains several results in this direction. In this paper, we use the
maximum number of facets of lattice-free sets in a family as a measure of its
complexity and precisely characterize the level of complexity of a family
required for constant factor approximations. As one of the main results, we
show that, for each natural number , a corner polyhedron with basic
integer variables and an arbitrary number of continuous non-basic variables is
approximated up to a constant factor by intersection cuts from lattice-free
sets with at most facets if and that no such approximation is
possible if . When the approximation factor is allowed to
depend on the denominator of the fractional vertex of the linear relaxation of
the corner polyhedron, we show that the threshold is versus .
The tools introduced for proving such results are of independent interest for
studying intersection cuts
Integrality gaps of integer knapsack problems
We obtain optimal lower and upper bounds for the (additive) integrality
gaps of integer knapsack problems. In a randomised setting, we show that the integrality
gap of a âtypicalâ knapsack problem is drastically smaller than the integrality
gap that occurs in a worst case scenario
Verifying integer programming results
Software for mixed-integer linear programming can return incorrect results for a number of reasons, one being the use of inexact floating-point arithmetic. Even solvers that employ exact arithmetic may suffer from programming or algorithmic errors, motivating the desire for a way to produce independently verifiable certificates of claimed results. Due to the complex nature of state-of-the-art MIP solution algorithms, the ideal form of such a certificate is not entirely clear. This paper proposes such a certificate format designed with simplicity in mind, which is composed of a list of statements that can be sequentially verified using a limited number of inference rules. We present a supplementary verification tool for compressing and checking these certificates independently of how they were created. We report computational results on a selection of MIP instances from the literature. To this end, we have extended the exact rational version of the MIP solver SCIP to produce such certificates
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