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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 [59] . 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: $X=P\cap {\mathbb{Z}}^n\times {ℝ}^p$ where $P=\{x\in {ℝ}^{n+p}:Ax\le b\}$ 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 $conv\left(X\right)=\{x\in {ℝ}^{n+p}:Cx\le d\}$ 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 $\pi x\ge {\pi }_0$ 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 $Cx\le d$ 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.