6,930 research outputs found
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
Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting
Mixed-integer mathematical programs are among the most commonly used models
for a wide set of problems in Operations Research and related fields. However,
there is still very little known about what can be expressed by small
mixed-integer programs. In particular, prior to this work, it was open whether
some classical problems, like the minimum odd-cut problem, can be expressed by
a compact mixed-integer program with few (even constantly many) integer
variables. This is in stark contrast to linear formulations, where recent
breakthroughs in the field of extended formulations have shown that many
polytopes associated to classical combinatorial optimization problems do not
even admit approximate extended formulations of sub-exponential size.
We provide a general framework for lifting inapproximability results of
extended formulations to the setting of mixed-integer extended formulations,
and obtain almost tight lower bounds on the number of integer variables needed
to describe a variety of classical combinatorial optimization problems. Among
the implications we obtain, we show that any mixed-integer extended formulation
of sub-exponential size for the matching polytope, cut polytope, traveling
salesman polytope or dominant of the odd-cut polytope, needs many integer variables, where is the number of vertices of the
underlying graph. Conversely, the above-mentioned polyhedra admit
polynomial-size mixed-integer formulations with only or (for the traveling salesman polytope) many integer variables.
Our results build upon a new decomposition technique that, for any convex set
, allows for approximating any mixed-integer description of by the
intersection of with the union of a small number of affine subspaces.Comment: A conference version of this paper will be presented at SODA 201
Intermediate integer programming representations using value disjunctions
We introduce a general technique to create an extended formulation of a
mixed-integer program. We classify the integer variables into blocks, each of
which generates a finite set of vector values. The extended formulation is
constructed by creating a new binary variable for each generated value. Initial
experiments show that the extended formulation can have a more compact complete
description than the original formulation.
We prove that, using this reformulation technique, the facet description
decomposes into one ``linking polyhedron'' per block and the ``aggregated
polyhedron''. Each of these polyhedra can be analyzed separately. For the case
of identical coefficients in a block, we provide a complete description of the
linking polyhedron and a polynomial-time separation algorithm. Applied to the
knapsack with a fixed number of distinct coefficients, this theorem provides a
complete description in an extended space with a polynomial number of
variables.Comment: 26 pages, 5 figure
A note on the split rank of intersection cuts
In this note, we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts. As a first step of this argument, a polyhedral subset of the lattice-free convex set that is used to generate the intersection cut is constructed. We call this subset the restricted lattice-free set. It is then shown that ! log 2(l)mixed integer programming, split rank, intersection cuts.
Constrained infinite group relaxations of MIPs
Recently minimal and extreme inequalities for continuous group relaxations of general mixed integer sets have been characterized. In this paper, we consider a stronger relaxation of general mixed integer sets by allowing constraints, such as bounds, on the free integer variables in the continuous group relaxation. We generalize a number of results for the continuous infinite group relaxation to this stronger relaxation and characterize the extreme inequalities when there are two integer variables.
A Framework for Globally Optimizing Mixed-Integer Signomial Programs
Mixed-integer signomial optimization problems have broad applicability in engineering. Extending the Global Mixed-Integer Quadratic Optimizer, GloMIQO (Misener, Floudas in J. Glob. Optim., 2012. doi:10.1007/s10898-012-9874-7), this manuscript documents a computational framework for deterministically addressing mixed-integer signomial optimization problems to ε-global optimality. This framework generalizes the GloMIQO strategies of (1) reformulating user input, (2) detecting special mathematical structure, and (3) globally optimizing the mixed-integer nonconvex program. Novel contributions of this paper include: flattening an expression tree towards term-based data structures; introducing additional nonconvex terms to interlink expressions; integrating a dynamic implementation of the reformulation-linearization technique into the branch-and-cut tree; designing term-based underestimators that specialize relaxation strategies according to variable bounds in the current tree node. Computational results are presented along with comparison of the computational framework to several state-of-the-art solvers. © 2013 Springer Science+Business Media New York
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