108 research outputs found
On the complexity of nonlinear mixed-integer optimization
This is a survey on the computational complexity of nonlinear mixed-integer
optimization. It highlights a selection of important topics, ranging from
incomputability results that arise from number theory and logic, to recently
obtained fully polynomial time approximation schemes in fixed dimension, and to
strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear
Optimization, IMA Volumes, Springer-Verla
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
Computing parametric rational generating functions with a primal Barvinok algorithm
Computations with Barvinok's short rational generating functions are
traditionally being performed in the dual space, to avoid the combinatorial
complexity of inclusion--exclusion formulas for the intersecting proper faces
of cones. We prove that, on the level of indicator functions of polyhedra,
there is no need for using inclusion--exclusion formulas to account for
boundary effects: All linear identities in the space of indicator functions can
be purely expressed using half-open variants of the full-dimensional polyhedra
in the identity. This gives rise to a practically efficient, parametric
Barvinok algorithm in the primal space.Comment: 16 pages, 1 figure; v2: Minor corrections, new example and summary of
algorithm; submitted to journa
A primal Barvinok algorithm based on irrational decompositions
We introduce variants of Barvinok's algorithm for counting lattice points in
polyhedra. The new algorithms are based on irrational signed decomposition in
the primal space and the construction of rational generating functions for
cones with low index. We give computational results that show that the new
algorithms are faster than the existing algorithms by a large factor.Comment: v3: New all-primal algorithm. v4: Extended introduction, updated
computational results. To appear in SIAM Journal on Discrete Mathematic
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
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
Computation of Atomic Fibers of Z-Linear Maps
For given matrix , the set
describes the preimage or fiber of under the -linear map
, . The fiber is called atomic, if
implies or . In this paper we present a
novel algorithm to compute such atomic fibers. An algorithmic solution to
appearing subproblems, computational examples and applications are included as
well.Comment: 27 page
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