25 research outputs found
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.
When Lift-and-Project Cuts are Different
In this paper, we present a method to determine if a lift-and-project cut for
a mixed-integer linear program is irregular, in which case the cut is not
equivalent to any intersection cut from the bases of the linear relaxation.
This is an important question due to the intense research activity for the past
decade on cuts from multiple rows of simplex tableau as well as on
lift-and-project cuts from non-split disjunctions. While it is known since
Balas and Perregaard (2003) that lift-and-project cuts from split disjunctions
are always equivalent to intersection cuts and consequently to such multi-row
cuts, Balas and Kis (2016) have recently shown that there is a necessary and
sufficient condition in the case of arbitrary disjunctions: a lift-and-project
cut is regular if, and only if, it corresponds to a regular basic solution of
the Cut Generating Linear Program (CGLP). This paper has four contributions.
First, we state a result that simplifies the verification of regularity for
basic CGLP solutions from Balas and Kis (2016). Second, we provide a
mixed-integer formulation that checks whether there is a regular CGLP solution
for a given cut that is regular in a broader sense, which also encompasses
irregular cuts that are implied by the regular cut closure. Third, we describe
a numerical procedure based on such formulation that identifies irregular
lift-and-project cuts. Finally, we use this method to evaluate how often
lift-and-project cuts from simple -branch split disjunctions are irregular,
and thus not equivalent to multi-row cuts, on 74 instances of the MIPLIB
benchmarks.Comment: INFORMS Journal on Computing (to appear
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