Implementation of Cutting Plane Separators for Mixed Integer ... - ZIB

Chapter 3
Cutting Plane Separator for the
Class of C-MIR Inequalities
In this chapter, we investigate the implementation of an efficient cutting plane separator
which generates general cutting planes for MIPs. More precisely, we separate
the class of complemented mixed integer rounding inequalities (c-MIR inequalities).
3.1 Introduction
Cutting planes which do not require or exploit any knowledge about the underlying
problem are called general cutting planes. Several families of general cutting planes
for MIPs have been proposed in the literature over the last fifty years, including
Gomory mixed integer cuts (GMI cuts) [28], intersection cuts [7], disjunction cuts
[9], split cuts [16], and mixed integer rounding cuts (MIR cuts) [48]. It is well-known
that many of these inequalities, such as GMI inequalities and MIR inequalities, are
equivalent (see [19, 20]).
In contrast to cutting plane separators that are based on the polyhedral analysis
of the problem formulation, for decades, general cutting planes were considered to be
impractical for solving large size instances of MIPs. However, in the 1990s, motivated
by their results for implementing lift-and-project cuts [10] for BMIPs, Balas, Ceria,
Cornuéjols, and Natraj [11] revisited GMI cuts and demonstrated their practical
usefulness within a linear programming based branch-and-cut algorithm (see also
[18]).
Marchand and Wolsey [39, 42] also pursued the idea of using general cutting
planes for MIPs. They investigated the practical usefulness of MIR cuts. However,
their work is based on the idea that even general cuts should be based, if possible,
on problem structure, and that this problem structure is to be found in the original
problem matrix (see [39]). They considered mixed knapsack sets which are obtained
by taking linear combinations of constraints (transformed into equality form) of the
original MIP and by substituting bounds imposed on the real variables. For these
sets, they introduced the class of c-MIR inequalities, which are MIR inequalities
derived for the mixed knapsack sets after scaling the mixed knapsack constraint
and complementing some of the integer variables. This idea was motivated by the
observation that several strong valid inequalities based on specific problem structure,
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