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Zero-one IP problems: Polyhedral descriptions & cutting plane procedures
A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie
Primary Facets Of Order Polytopes
Mixture models on order relations play a central role in recent
investigations of transitivity in binary choice data. In such a model, the
vectors of choice probabilities are the convex combinations of the
characteristic vectors of all order relations of a chosen type. The five
prominent types of order relations are linear orders, weak orders, semiorders,
interval orders and partial orders. For each of them, the problem of finding a
complete, workable characterization of the vectors of probabilities is
crucial---but it is reputably inaccessible. Under a geometric reformulation,
the problem asks for a linear description of a convex polytope whose vertices
are known. As for any convex polytope, a shortest linear description comprises
one linear inequality per facet. Getting all of the facet-defining inequalities
of any of the five order polytopes seems presently out of reach. Here we search
for the facet-defining inequalities which we call primary because their
coefficients take only the values -1, 0 or 1. We provide a classification of
all primary, facet-defining inequalities of three of the five order polytopes.
Moreover, we elaborate on the intricacy of the primary facet-defining
inequalities of the linear order and the weak order polytopes
Looking for symmetric Bell inequalities
Finding all Bell inequalities for a given number of parties, measurement
settings, and measurement outcomes is in general a computationally hard task.
We show that all Bell inequalities which are symmetric under the exchange of
parties can be found by examining a symmetrized polytope which is simpler than
the full Bell polytope. As an illustration of our method, we generate 238885
new Bell inequalities and 1085 new Svetlichny inequalities. We find, in
particular, facet inequalities for Bell experiments involving two parties and
two measurement settings that are not of the
Collins-Gisin-Linden-Massar-Popescu type.Comment: Joined the associated website as an ancillary file, 17 pages, 1
figure, 1 tabl
Lifting Bell inequalities
A Bell inequality defined for a specific experimental configuration can always be extended to a situation involving more observers, measurement settings, or measurement outcomes. In this article, such "liftings" of Bell inequalities are studied. It is shown that if the original inequality defines a facet of the polytope of local joint outcome probabilities then the lifted one also defines a facet of the more complex polytope
Facets of a mixed-integer bilinear covering set with bounds on variables
We derive a closed form description of the convex hull of mixed-integer
bilinear covering set with bounds on the integer variables. This convex hull
description is determined by considering some orthogonal disjunctive sets
defined in a certain way. This description does not introduce any new
variables, but consists of exponentially many inequalities. An extended
formulation with a few extra variables and much smaller number of constraints
is presented. We also derive a linear time separation algorithm for finding the
facet defining inequalities of this convex hull. We study the effectiveness of
the new inequalities and the extended formulation using some examples
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