Laboratoire de Génie Informatique et d’Automatique de l’Artois

We study a general optimization problem in which the coefficients in the objective are uncertain, focusing on cases of severe uncertainty, i.e., when a single probability measure is inadequate as an uncertainty model. Therefore, we use more general frameworks, namely belief functions and lower probabilities (capacities), which enable the application of common criteria in the literature to select non-dominated solutions. When the uncertainty is modeled by a belief function whose focal sets are Cartesian productsf compact sets, we provide characterizations of the non-dominated solutions of the generalized Hurwicz, strong dominance, weak dominance, maximality, and E-admissibility criteria. When the uncertainty is modeled by a lower probability on a finite frame, we provide characterizations of the non-dominated solutions of maximality and E-admissibility. All these characterizations correspond to established notions in optimization. Furthermore, they make it possible to derive several interesting results, notably the efficiency of finding non-dominated solutions or the equivalence of maximality and E-admissibility, in many situations. Lastly, for the generalized min-max regret criterion under these two uncertainty models, we develop approximation methods extending the well-known midpoint methods used in robust min-max regret optimization with interval and discrete data.