3,057 research outputs found
On optimizing over lift-and-project closures
The lift-and-project closure is the relaxation obtained by computing all
lift-and-project cuts from the initial formulation of a mixed integer linear
program or equivalently by computing all mixed integer Gomory cuts read from
all tableau's corresponding to feasible and infeasible bases. In this paper, we
present an algorithm for approximating the value of the lift-and-project
closure. The originality of our method is that it is based on a very simple cut
generation linear programming problem which is obtained from the original
linear relaxation by simply modifying the bounds on the variables and
constraints. This separation LP can also be seen as the dual of the cut
generation LP used in disjunctive programming procedures with a particular
normalization. We study some properties of this separation LP in particular
relating it to the equivalence between lift-and-project cuts and Gomory cuts
shown by Balas and Perregaard. Finally, we present some computational
experiments and comparisons with recent related works
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
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.
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
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