123 research outputs found
Split rank of triangle and quadrilateral inequalities
A simple relaxation of two rows of a simplex tableau is a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. Recently Andersen et al. [2] and Cornu´ejols and Margot [13] showed that the facet-defining inequalities of this set are either split cuts or intersection cuts obtained from lattice-free triangles and quadrilaterals. Through a result by Cook et al. [12], it is known that one particular class of facet- defining triangle inequality does not have a finite split rank. In this paper, we show that all other facet-defining triangle and quadrilateral inequalities have finite split rank. The proof is constructive and given a facet-defining triangle or quadrilateral inequality we present an explicit sequence of split inequalities that can be used to generate it.mixed integer programs, split rank, group relaxations
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.
Some lower bounds on sparse outer approximations of polytopes
Motivated by the need to better understand the properties of sparse
cutting-planes used in mixed integer programming solvers, the paper [2] studied
the idealized problem of how well a polytope is approximated by the use of
sparse valid inequalities. As an extension to this work, we study the following
less idealized questions in this paper: (1) Are there integer programs, such
that sparse inequalities do not approximate the integer hull well even when
added to a linear programming relaxation? (2) Are there polytopes, where the
quality of approximation by sparse inequalities cannot be significantly
improved by adding a budgeted number of arbitrary (possibly dense) valid
inequalities? (3) Are there polytopes that are difficult to approximate under
every rotation? (4) Are there polytopes that are difficult to approximate in
all directions using sparse inequalities? We answer each of the above questions
in the positive
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