Section: New Results

Optimization

Semidefinite programming and combinatorial optimization

Participant : Jérôme Malick.

We have worked with Frederic Roupin (Prof. at Paris XIII) and Nathan Krislock (Assistant Prof. at North Illinois University, USA) on the use of semidefinite programming to solve combinatorial optimization problems to optimality.

We proposed a new family of semidefinite bounds for 0-1 quadratic problems with linear or quadratic constraints [65] . We have embedded the new bounds within branch-and-bound algorithms to solve 2 standard combinatorial optimization problems to optimality.

• Max-cut. We developed [26] an improved bounding procedure obtained by reducing two key parameters (the target level of accuracy and the stopping tolerance of the inner Quasi-Newton engine) to zero, and iteratively adding triangle inequality cuts. We also precisely analyzed its theoretical convergence properties. We show that our method outperform the state-of-the-art solver ( [66] ) on the large test-problems.

• Heaviest k-subgraph problems. Adapting the techniques we developped for the max-cut problem, we have proposed in [60] an algorithm able to solve exactly k-cluster instances of size 160. In practice, our method works particularly fine on the most difficult instances (with a large number of vertices, small density and small k).

We have also been working on a generic online semidefinite-based solver for binary quadratic problems using the generality of [65] . Finally, a first web interface for our solvers and our data sets are available online at http://lipn.univ-paris13.fr/BiqCrunch/ .

Quadratic stabilization of Benders decomposition

Participants : Jérôme Malick, Sofia Zaourar.

The Benders decomposition, a fundamental method in operation research, is known to have the inherent instability of cutting plane-based methods. The PhD thesis of Sofia Zaourar proposes a algorithmic improvement of the method inspired from the level-bundle methods of nonsmooth optimisation. We illustrate the interest of the stabilization on two classical network problems: network design problems and hub location problems. We also prove that the stabilized Benders method have the same theoretical convergence properties as the usual Benders method. An article about this research was submitted this summer.