4,359 research outputs found
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
Complexity of optimizing over the integers
In the first part of this paper, we present a unified framework for analyzing
the algorithmic complexity of any optimization problem, whether it be
continuous or discrete in nature. This helps to formalize notions like "input",
"size" and "complexity" in the context of general mathematical optimization,
avoiding context dependent definitions which is one of the sources of
difference in the treatment of complexity within continuous and discrete
optimization. In the second part of the paper, we employ the language developed
in the first part to study information theoretic and algorithmic complexity of
{\em mixed-integer convex optimization}, which contains as a special case
continuous convex optimization on the one hand and pure integer optimization on
the other. We strive for the maximum possible generality in our exposition.
We hope that this paper contains material that both continuous optimizers and
discrete optimizers find new and interesting, even though almost all of the
material presented is common knowledge in one or the other community. We see
the main merit of this paper as bringing together all of this information under
one unifying umbrella with the hope that this will act as yet another catalyst
for more interaction across the continuous-discrete divide. In fact, our
motivation behind Part I of the paper is to provide a common language for both
communities
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