679 research outputs found
Split rank of triangle and quadrilateral inequalities
A simple relaxation of two rows of a simplex tableau is a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. Recently Andersen et al. [2] and Cornu´ejols and Margot [13] showed that the facet-defining inequalities of this set are either split cuts or intersection cuts obtained from lattice-free triangles and quadrilaterals. Through a result by Cook et al. [12], it is known that one particular class of facet- defining triangle inequality does not have a finite split rank. In this paper, we show that all other facet-defining triangle and quadrilateral inequalities have finite split rank. The proof is constructive and given a facet-defining triangle or quadrilateral inequality we present an explicit sequence of split inequalities that can be used to generate it.mixed integer programs, split rank, group relaxations
Split rank of triangle and quadrilateral inequalities
A simple relaxation of two rows of a simplex tableau is a mixed integer set
consisting of two equations with two free integer variables and non-negative
continuous variables. Recently Andersen, Louveaux, Weismantel and Wolsey (2007)
and Cornuejols and Margot (2008) showed that the facet-defining inequalities of
this set are either split cuts or intersection cuts obtained from lattice-free
triangles and quadrilaterals. Through a result by Cook, Kannan and Schrijver
(1990), it is known that one particular class of facet-defining triangle
inequality does not have a finite split rank. In this paper, we show that all
other facet-defining triangle and quadrilateral inequalities have a finite
split-rank. The proof is constructive and given a facet-defining triangle or
quadrilateral inequality we present an explicit sequence of split inequalities
that can be used to generate it.Comment: 39 pages and 13 figure
The Triangle Closure is a Polyhedron
Recently, cutting planes derived from maximal lattice-free convex sets have
been studied intensively by the integer programming community. An important
question in this research area has been to decide whether the closures
associated with certain families of lattice-free sets are polyhedra. For a long
time, the only result known was the celebrated theorem of Cook, Kannan and
Schrijver who showed that the split closure is a polyhedron. Although some
fairly general results were obtained by Andersen, Louveaux and Weismantel [ An
analysis of mixed integer linear sets based on lattice point free convex sets,
Math. Oper. Res. 35 (2010), 233--256] and Averkov [On finitely generated
closures in the theory of cutting planes, Discrete Optimization 9 (2012), no.
4, 209--215], some basic questions have remained unresolved. For example,
maximal lattice-free triangles are the natural family to study beyond the
family of splits and it has been a standing open problem to decide whether the
triangle closure is a polyhedron. In this paper, we show that when the number
of integer variables the triangle closure is indeed a polyhedron and its
number of facets can be bounded by a polynomial in the size of the input data.
The techniques of this proof are also used to give a refinement of necessary
conditions for valid inequalities being facet-defining due to Cornu\'ejols and
Margot [On the facets of mixed integer programs with two integer variables and
two constraints, Mathematical Programming 120 (2009), 429--456] and obtain
polynomial complexity results about the mixed integer hull.Comment: 39 pages; made self-contained by merging material from
arXiv:1107.5068v
Integral Farkas type lemmas for systems with equalities and inequalities
A central result in the theory of integer optimization states that a system of linear diophantine equations Ax = b has no integral solution if and only if there exists a vector in the dual lattice, y T A integral such that y T b is fractional. We extend this result to systems that both have equations and inequalities {Ax = b, Cx d}. We show that a certificate of integral infeasibility is a linear system with rank(C) variables containing no integral point.
Spherical metrics with conical singularities on a 2-sphere: angle constraints
In this article we give a criterion for the existence of a metric of
curvature on a -sphere with conical singularities of prescribed
angles and non-coaxial holonomy. Such a
necessary and sufficient condition is expressed in terms of linear inequalities
in .Comment: 38 pages, 17 figure
An algorithm for the separation of two-row cuts
peer reviewedWe consider the question of finding deep cuts from a model with two rows of the type PI = {(x,s) ∈ Z2 ×Rn+ : x = f +Rs}. To do that, we show how to reduce the complexity of setting up the polar of conv(PI ) from a quadratic number of integer hull computations to a linear number of integer hull computations. Furthermore, we present an algorithm that avoids computing all integer hulls. A polynomial running time is not guaranteed but computational results show that the algorithm runs quickly in practice
Relaxations of mixed integer sets from lattice-free polyhedra
This paper gives an introduction to a recently established link between the geometry of numbers and mixed integer optimization. The main focus is to provide a review of families of lattice-free polyhedra and their use in a disjunctive programming approach. The use of lattice-free polyhedra in the context of deriving and explaining cutting planes for mixed integer programs is not only mathematically interesting, but it leads to some fundamental new discoveries, such as an understanding under which conditions cutting planes algorithms converge finitel
GPU-accelerated discontinuous Galerkin methods on hybrid meshes
We present a time-explicit discontinuous Galerkin (DG) solver for the
time-domain acoustic wave equation on hybrid meshes containing vertex-mapped
hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable
formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto
(Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions
for hybrid meshes are derived by bounding the spectral radius of the DG
operator using order-dependent constants in trace and Markov inequalities.
Computational efficiency is achieved under a combination of element-specific
kernels (including new quadrature-free operators for the pyramid), multi-rate
timestepping, and acceleration using Graphics Processing Units.Comment: Submitted to CMAM
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