59,170 research outputs found
A characterization of maximal homogeneous-quadratic-free sets
The intersection cut framework was introduced by Balas in 1971 as a method
for generating cutting planes in integer optimization. In this framework, one
uses a full-dimensional convex -free set, where is the feasible region
of the integer program, to derive a cut separating from a non-integral
vertex of a linear relaxation of . Among all -free sets, it is the
inclusion-wise maximal ones that yield the strongest cuts. Recently, this
framework has been extended beyond the integer case in order to obtain cutting
planes in non-linear settings. In this work, we consider the specific setting
when is defined by a homogeneous quadratic inequality. In this
'quadratic-free' setting, every function , where is
the unit disk in , generates a representation of a quadratic-free
set. While not every generates a maximal quadratic free set, it is the
case that every full-dimensional maximal quadratic free set is generated by
some . Our main result shows that the corresponding quadratic-free set
is full-dimensional and maximal if and only if is non-expansive and
satisfies a technical condition. This result yields a broader class of maximal
-free sets than previously known. Our result stems from a new
characterization of maximal -free sets (for general beyond the quadratic
setting) based on sequences that 'expose' inequalities defining the -free
set
Cut-generating functions and S-free sets
International audienceWe consider the separation problem for sets X that are pre-images of a given set S by a linear mapping. Classical examples occur in integer programming, as well as in other optimization problems such as complementarity. One would like to generate valid inequalities that cut off some point not lying in X, without reference to the linear mapping. To this aim, we introduce a concept: cut-generating functions (cgf) and we develop a formal theory for them, largely based on convex analysis. They are intimately related to S-free sets and we study this relation, disclosing several definitions for minimal cgf's and maximal S-free sets. Our work unifies and puts in perspective a number of existing works on S-free sets; in particular, we show how cgf's recover the celebrated Gomory cuts
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
Optimality certificates for convex minimization and Helly numbers
We consider the problem of minimizing a convex function over a subset of R^n
that is not necessarily convex (minimization of a convex function over the
integer points in a polytope is a special case). We define a family of duals
for this problem and show that, under some natural conditions, strong duality
holds for a dual problem in this family that is more restrictive than
previously considered duals.Comment: 5 page
Discrete convexity and unimodularity. I
In this paper we develop a theory of convexity for a free Abelian group M
(the lattice of integer points), which we call theory of discrete convexity. We
characterize those subsets X of the group M that could be call "convex". One
property seems indisputable: X should coincide with the set of all integer
points of its convex hull co(X) (in the ambient vector space V). However, this
is a first approximation to a proper discrete convexity, because such
non-intersecting sets need not be separated by a hyperplane. This issue is
closely related to the question when the intersection of two integer polyhedra
is an integer polyhedron. We show that unimodular systems (or more generally,
pure systems) are in one-to-one correspondence with the classes of discrete
convexity. For example, the well-known class of g-polymatroids corresponds to
the class of discrete convexity associated to the unimodular system A_n:={\pm
e_i, e_i-ej} in Z^n.Comment: 26 pages, Late
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