## Section: Scientific Foundations

### Decomposition and reformulation approaches

An hierarchical approach to tackle complex combinatorial problems
consists in considering separately different substructures
(subproblems).
If one is able to implement relatively
efficient optimization on the substructures, this can be
exploited to reformulate the global problem as a
selection of specific subproblem solutions that
together form a global solution. If the subproblems correspond to
subset of constraints in the MIP formulation, this leads to
Dantzig-Wolfe decomposition. If it corresponds to isolating a subset of
decision variables, this leads to
Bender's decomposition. Both lead to extended formulations of the
problem with either a huge number of variables or
constraints. Dantzig-Wolfe approach requires specific algorithmic approaches to generate subproblem solutions and associated global decision variables dynamically in
the course of the optimization. This procedure is known as *column
generation*, while its combination with branch-and-bound enumeration
is called, *branch-and-price*. Alternatively, in Bender's
approach, when dealing with
exponentially many constraints in the reformulation, *cutting
plane procedures* defined in the previous section reveal to be
powerful. When optimization on a substructure is (relatively) easy,
there often exists a tight reformulation of this substructure typically in an extended variable space.
This gives rise powerful reformulation of the global problem,
although it might be impractical given its size
(typically pseudo-polynomial). It can be possible to project
(part of) the extended formulation in a smaller dimensional space if not the
original variable space to bring polyhedral insight (cuts derived
through polyhedral studies can often be recovered through such projections).