Section: New Results

Optimization

We developped a new approach exploiting the hidden underlying lower-dimension structure of this big data. We proposed a new family of algorithms (of the type coordinate results, or conditional gadient), whose iterations have an algorithmic complexity lower that an order compared to standard methods. For example, applied to learning problems with trace-norm penalization, our algorithm [26] exploit the atomic decomposition of the norm and compute only an approximate largest singular vector pair (instead of the whole singular value decomposition). Promising results [27] have been obtained on the image database Imagenet, where we show significant improvements compare to the state-of-the-art approaches (one-vs-rest strategies).

We proposed a new family of semidefinite bounds for 0-1 quadratic problems with linear or quadratic constraints [50] . The paradigm is to trade computing time for a (small) deterioration of the quality of the usual semidefinite bounds, in view of enhancing this efficiency in exact resolution schemes. Extensive numerical comparisons et tests showed the superior quality of our bounds, when embedded within branch-and-bound shemes, on standard test-problems (unconstrained 0-1 quadratic problems, heaviest k-subgraph problems, and graph bisection problems).

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

• Max-cut. We developed [34] 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 ( [52] ) on the large test-problems.

• Heaviest k-subgraph problems. Our previous work [51] takes advantage of the new bounds in their basic form to prune very well in the search tree. Its performances are then comparable with the best method (based on convex quadratic relaxation using CPLEX as an engine). Adapting and incorporating the tehniques we developped for the max-cut problem, we propose in [35] an big improveement 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).

Finally, we have worked on making our data sets available online together with a web interface for our solvers. We have also started working on a generic online semidefinite-based solver for binary quadratic problems using the generality of [50] . All this is publicly available on line at http://lipn.univ-paris13.fr/BiqCrunch/ .

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 [48] . However, the computed optimal dual variables show a noisy and unstable behaviour, that could prevent their use as price indicator. We propose a simple and controllable way to stabilize the dual optimal solutions, by penalizing the total variation of the prices [36] . Our illustrations on the daily electricity production optimization of EDF show a strinking stabilization at a negligible cost.