Cuts from two rows of a simplex tableau

approach is to separate a cut from a subspace defined by the two basic integer variables and a small
number of non-basic continuous variables. The resulting cut is then lifted into the space of the
remaining variables.

Using this approach, we obtain facet defining inequalities for the mixed integer set defined by two rows
of a simplex tableau. The inequalities are given by a closed form formula.
We demonstrate that this class of cuts includes cuts that are not split cuts.
In particular, we consider an example of Cook, Kannan and Schrijver for
which the split closure is different than the mixed integer hull. We demonstrate
that the missing inequality in this example can be obtained from the two rows
of the simplex tableau that define the only fractional solution.

We provide two different viewpoints. One is geometric, and the other is
algebraic. Geometrically, we view the mixed integer set as defined by
a cone with an apex and a number of extreme rays, i.e., as a corner polyhedron. The
geometric view leads to a correspondence between lifted inequalities and two-dimensional polygons.
Algebraically, we view the mixed integer set as a mixed integer program with
two inequalities, and cuts are derived from Mixed Integer Rounding (MIR) and
from disjunctive arguments.