392 research outputs found
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
An analysis of mixed integer linear sets based on lattice point free convex sets
Split cuts are cutting planes for mixed integer programs whose validity is
derived from maximal lattice point free polyhedra of the form called split sets. The set obtained by adding all
split cuts is called the split closure, and the split closure is known to be a
polyhedron. A split set has max-facet-width equal to one in the sense that
. In this paper
we consider using general lattice point free rational polyhedra to derive valid
cuts for mixed integer linear sets. We say that lattice point free polyhedra
with max-facet-width equal to have width size . A split cut of width
size is then a valid inequality whose validity follows from a lattice point
free rational polyhedron of width size . The -th split closure is the set
obtained by adding all valid inequalities of width size at most . Our main
result is a sufficient condition for the addition of a family of rational
inequalities to result in a polyhedral relaxation. We then show that a
corollary is that the -th split closure is a polyhedron. Given this result,
a natural question is which width size is required to design a finite
cutting plane proof for the validity of an inequality. Specifically, for this
value , a finite cutting plane proof exists that uses lattice point free
rational polyhedra of width size at most , but no finite cutting plane
proof that only uses lattice point free rational polyhedra of width size
smaller than . We characterize based on the faces of the linear
relaxation
Reverse Chv\'atal-Gomory rank
We introduce the reverse Chv\'atal-Gomory rank r*(P) of an integral
polyhedron P, defined as the supremum of the Chv\'atal-Gomory ranks of all
rational polyhedra whose integer hull is P. A well-known example in dimension
two shows that there exist integral polytopes P with r*(P) equal to infinity.
We provide a geometric characterization of polyhedra with this property in
general dimension, and investigate upper bounds on r*(P) when this value is
finite.Comment: 21 pages, 4 figure
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.
On Minimal Valid Inequalities for Mixed Integer Conic Programs
We study disjunctive conic sets involving a general regular (closed, convex,
full dimensional, and pointed) cone K such as the nonnegative orthant, the
Lorentz cone or the positive semidefinite cone. In a unified framework, we
introduce K-minimal inequalities and show that under mild assumptions, these
inequalities together with the trivial cone-implied inequalities are sufficient
to describe the convex hull. We study the properties of K-minimal inequalities
by establishing algebraic necessary conditions for an inequality to be
K-minimal. This characterization leads to a broader algebraically defined class
of K- sublinear inequalities. We establish a close connection between
K-sublinear inequalities and the support functions of sets with a particular
structure. This connection results in practical ways of showing that a given
inequality is K-sublinear and K-minimal.
Our framework generalizes some of the results from the mixed integer linear
case. It is well known that the minimal inequalities for mixed integer linear
programs are generated by sublinear (positively homogeneous, subadditive and
convex) functions that are also piecewise linear. This result is easily
recovered by our analysis. Whenever possible we highlight the connections to
the existing literature. However, our study unveils that such a cut generating
function view treating the data associated with each individual variable
independently is not possible in the case of general cones other than
nonnegative orthant, even when the cone involved is the Lorentz cone
- …