Représentations discrètes de l'ensemble des points non dominés pour des problèmes d'optimisation multi-objectifs

The goal of this thesis is to propose new general methods to get around the intractability of multi-objective optimization problems.First, we try to give some insight on this intractability by determining an, easily computable, upper bound on the number of nondominated points, knowing the number of values taken on each criterion. Then, we are interested in producingsome discrete and tractable representations of the set of nondominated points for each instance of multi-objective optimization problems. These representations must satisfy some conditions of coverage, i.e. providing a good approximation, cardinality, i.e. it does not contain too many points, and if possible spacing, i.e. it does not include any redundancies. Starting from works aiming to produce ε-Pareto sets of small size, we first propose a direct extension of these works then we focus our research on ε-Pareto sets satisfying an additional condition of stability. Formally, we consider special ε-Pareto sets, called (ε, ε′)-kernels, which satisfy a property of stability related to ε′. We give some general results on (ε, ε′)-kernels and propose some polynomial time algorithms that produce small (ε, ε′)-kernels for the bicriteria case and we give some negative results for the tricriteria case and beyond.