## Section: Research Program

### Solving problems with complex structures

Standard solution methods developed for CS problems solve independent
sub-problems associated with each type of variables without explicitly
integrating their interactions or integrating them iteratively in a
heuristic way. However these subproblems are intrinsically linked and
should be addressed jointly. In *mathematical* *optimization*
a classical approach is to approximate the convex hull of the integer
solutions of the model by its linear relaxation. The main solution
methods are (1) polyhedral solution methods which strengthen this linear
relaxation by adding valid inequalities, (2) decomposition solution
methods (Dantzig Wolfe, Lagrangian Relaxation, Benders decomposition)
which aim to obtain a better
approximation and solve it by generating extreme points/rays. Main
challenges are (1) the analysis of the strength of the cuts and their
separations for polyhedral solution methods, (2) the decomposition
schemes and (3) the extreme points/rays generations for the
decomposition solution methods.

The main difficulty in solving *bilevel problems* is due to their
non convexity and non differentiability. Even linear bilevel programs,
where all functions involved are affine, are computationally challenging
despite their apparent simplicity. Up to now, much research has been devoted to
bilevel problems with linear or convex follower problems. In this case, the problem can be reformulated as a
single-level program involving complementarity constraints, exemplifying
the dual nature, continuous and combinatorial, of bilevel programs.