## Section: Scientific Foundations

### Polyhedral approaches for MIP

Adding valid inequalities to the polyhedral description of an MIP
allows one to improve the resulting LP bound and hence to better
prune the enumeration tree. In a cutting plane procedure, one
attempt to identify valid inequalities that are violated by the LP
solution of the current formulation and adds them to the
formulation. This can be done at each node of the branch-and-bound
tree giving rise to a so-called *branch-and-cut algorithm*
[69] . The goal is to reduce the resolution of an integer program to
that of a linear program by deriving a linear description of the
convex hull of the feasible solutions. Polyhedral theory tells us that
if $X$ is a mixed integer program: $X=P\cap {\mathbb{Z}}^{n}\times {\mathbb{R}}^{p}$ where $P=\{x\in {\mathbb{R}}^{n+p}:Ax\le b\}$ with matrix $(A,b)\in {\mathbb{Q}}^{m\times (n+p+1)}$, then
$conv\left(X\right)$ is a polyhedron that can be described in terms of linear
constraints,
i.e. it writes as $conv\left(X\right)=\{x\in {\mathbb{R}}^{n+p}:C\phantom{\rule{0.222222em}{0ex}}x\le d\}$
for some matrix $(C,d)\in {\mathbb{Q}}^{{m}^{\text{'}}\times (n+p+1)}$ although the dimension ${m}^{\text{'}}$
is typically quite large. A fundamental result in this field is the equivalence of
complexity between solving the combinatorial optimization problem
$min\{cx:x\in X\}$ and
solving the *separation problem* over the associated polyhedron $conv\left(X\right)$: if $\tilde{x}\notin conv\left(X\right)$, find a linear inequality $\pi \phantom{\rule{0.222222em}{0ex}}x\ge {\pi}_{0}$
satisfied by all points in $conv\left(X\right)$ but violated by $\tilde{x}$.
Hence, for NP-hard problems, one can not hope to get a compact
description of $conv\left(X\right)$ nor a polynomial time exact separation
routine. Polyhedral studies focus on identifying some of the
inequalities that are involved in the polyhedral description of
$conv\left(X\right)$ and derive efficient *separation procedures* (cutting
plane generation). Only a subset of the inequalities $C\phantom{\rule{0.222222em}{0ex}}x\le d$
can offer a good approximation, that combined with a branch-and-bound
enumeration techniques permits to solve the problem. Using *cutting
plane algorithm* at each node of the branch-and-bound tree, gives rise
to the algorithm called *branch-and-cut*.