## Section: Research Program

### 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, the *cutting plane procedures* that we defined in the previous section are
well-suited tools. 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).