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

Chapter 1
Introduction
A mixed integer program (MIP) is an optimization problem where the objective is
to minimize or maximize a function of many variables subject to inequality and
equality constraints and integrality restrictions on some or all of the variables. The
integrality restrictions are the crucial point. On the one hand, they give us the
power to formulate many real-world problems as MIPs, but on the other hand, they
make MIPs hard to solve; MIP-solving is NP-hard.
One technique to solve MIPs is the general cutting plane method. The idea is to
drop the integrality restrictions and to solve the resulting relaxation of the MIP by
efficient methods from linear programming. The dropped integrality restrictions are
then reintroduced by cutting planes. Algorithms that generate cutting planes are
called cutting plane separators. The actual power of this technique arises when it is
combined with a linear programming based branch-and-bound algorithm, which is
another technique to solve MIPs. Most state-of-the-art MIP solvers are based on this
combination, also known as linear programming based branch-and-cut algorithm.
Scip [1], developed at the Zuse Institute Berlin by T. Achterberg, is such a
framework. It integrates constraint and mixed integer programming. The aim of
this thesis is to find an efficient implementation of different cutting plane separators
which embedded in Scip help to improve the overall performance of the solver.
In [15], it has been shown that cutting plane separators are one of the most
important features of MIP solvers. But of course, their effect depends on the way
they are integrated into the MIP solver, i.e., on the way in which they interact with
other features of the solver. In this thesis, we concentrate on the performance of the
cutting plane separators when they are used basically isolated. We suppose that this
is a good starting point for developing cutting plane separators which are effective
within MIP solvers. At the end of this thesis, we remove the isolated application of
our developed cutting plane separators and present computational results concerning
their impact on the overall performance of Scip.
We deal with six different cutting plane separators. For three of them, namely the
cutting plane separators for the class of GMI inequalities, the node packing problem,
and the class of implied bound inequalities, we give only a brief introduction to the
classes of valid inequalities which are separated and mention some of the algorithmic
aspects.
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