- Nonlinear optimization
- Global optimization
- Multiobjective optimization
- Constraint programming
- Interval analysis
Since my master and PhD thesis, I have been interested in the development of methods for solving numerical nonlinear optimization problems, and in particular in a multiobjective context. The different approaches I am interested in range from local search in black box optimization context to global Branch & Bound methods. The use of numerical constraint programming techniques and interval analysis is in general useful in this latter context.
Multiobjective optimization problems consist of finding so called Pareto optimal solutions, i.e. solutions that represent optimal trade-offs of the different objectives such that there is no other solutions that can improve all objective functions simultaneously. This field has gained a growing interest over last fifteen years as the need of a good compromise appears in many decisions, such as in engineering design. For solving such problems, it is important to develop methods that both diversifies the search (search globally in order to reach all the different set of compromising solutions) and intensifies the search (search locally in order to attain accurately such compromises).
I am willing to explore in more depth the interaction between local and global search methods for solving various nonlinear multiobjective optimization problems. Many methods from the literature are evolutionary algorithms. By themselves, those methods lack of good convergence properties towards (locally) Pareto-optimal solutions. Hence, recent approaches combine the stochastic incomplete global exploration of the evolutionary algorithm with local search. Continuation methods are of that kind of local search approaches that have been recently the subject of many interests including mine. They manage, starting from an initial (locally) Pareto optimal solution, to discover efficiently other connected Pareto candidates showing different compromises of the objectives. Currently, most continuation approaches are solely applied to unconstrained biobjective problems.
Behind the use of continuation methods also lies the problematic of representing the Pareto-optimal solutions as continuation allows to abstract their representation as manifolds (instead of the usual set of punctual solutions). Efficient management of such representation could be very helpful for solving multiobjective problems, but also for decision making.