4,605 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
Computational Experiments with Cross and Crooked Cross Cuts
In this paper, we study whether cuts obtained from two simplex tableau rows at a time can strengthen the bounds obtained by Gomory mixed-integer (GMI) cuts based on single tableau rows. We also study whether cross and crooked cross cuts, which generalize split cuts, can be separated in an effective manner for practical mixed-integer programs (MIPs) and can yield a nontrivial improvement over the bounds obtained by split cuts. We give positive answers to both these questions for MIPLIB 3.0 problems. Cross cuts are a special case of the t-branch split cuts studied by Li and Richard [Li Y, Richard J-PP (2008) Cook, Kannan and Schrijvers's example revisited. Discrete Optim. 5:724–734]. Split cuts are 1-branch split cuts, and cross cuts are 2-branch split cuts. Crooked cross cuts were introduced by Dash, Günlük, and Lodi [Dash S, Günlük O, Lodi A (2010) MIR closures of polyhedral sets. Math Programming 121:33–60] and were shown to dominate cross cuts by Dash, Günlük, and Molinaro [Dash S, Günlük O, Molinaro M (2012b) On the relative strength of different generalizations of split cuts. IBM Technical Report RC25326, IBM, Yorktown Heights, NY].United States. Office of Naval Research (Grant N000141110724
Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. I. The One-Dimensional Case
We give an algorithm for testing the extremality of minimal valid functions
for Gomory and Johnson's infinite group problem that are piecewise linear
(possibly discontinuous) with rational breakpoints. This is the first set of
necessary and sufficient conditions that can be tested algorithmically for
deciding extremality in this important class of minimal valid functions. We
also present an extreme function that is a piecewise linear function with some
irrational breakpoints, whose extremality follows from a new principle.Comment: 38 pages, 10 figure
Lift-and-project inequalities
The lift-and-project technique is a systematic way to generate valid inequalities
for a mixed binary program. The technique is interesting both on the theoretical and
on the practical point of view. On the theoretical side it allows one to construct the
inequality description of the convex hull of all mixed-{0,1}
solutions of a binary MIP in n repeated applications of the technique, where
n is the number of binary variables.
On the practical side, a variant of the method allows one to derive some cutting planes
from the simplex tableau rather efficiently
An algorithm for the separation of two-row cuts
peer reviewedWe consider the question of finding deep cuts from a model with two rows of the type PI = {(x,s) ∈ Z2 ×Rn+ : x = f +Rs}. To do that, we show how to reduce the complexity of setting up the polar of conv(PI ) from a quadratic number of integer hull computations to a linear number of integer hull computations. Furthermore, we present an algorithm that avoids computing all integer hulls. A polynomial running time is not guaranteed but computational results show that the algorithm runs quickly in practice
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