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 [61] . We have embedded the new bounds within branch-and-bound algorithms to solve 2 standard combinatorial optimization problems to optimality.

• Max-cut. We developed [60] 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 ( [62] ) on the large test-problems.

• Heaviest k-subgraph problems. Adapting the techniques we developped for the max-cut problem, we have proposed in [59] an big improvement of the first algorithm (up to 10 times faster). For the first time, we were 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 [61] . Finally, a first web interface for our solvers and our data sets are available online at http://lipn.univ-paris13.fr/BiqCrunch/ .

On computing marginal prices in electricity production

Participants : Jérôme Malick, Sofia Zaourar.

Unit-commitment optimization problems in electricity production are large-scale, nonconvex and heterogeneous, but they are decomposable by Lagrangian duality. Realistic modeling of technical production constraints makes the dual objective function computed inexactly though. An inexact version of the bundle method has been dedicated to tackle this difficulty [58] . We have worked on two projects related to solving dual unit-commitment problem by inexact bundle methods.

• Stabilization. We observed that the computed optimal dual variables show a noisy and unstable behaviour, that could prevent their use as price indicator. We have proposed a simple and controllable way to stabilize the dual optimal solutions, by penalizing the total variation of the prices [63] . Our illustrations on the daily electricity production optimization of EDF show a strinking stabilization at a negligible cost.

• Acceleration. We have worked with Welington Oliveira (IMPA, Brazil) on the acceleration of inexact bundle methods by taking advantage of cheap-to-get inexact information on the objective function which comes without any tighness guarantee though. We came up with a new family of bundle methods incorporating this coarse inexact information, to get better iterates. We have studied the convergence of these method and we have conducted numerical experimentation on unit-commitment problems and on two-stage linear problems show a subtantial gain in the overall computing time. This research is about to be released in a preprint in HAL