James LARROUY (LAMIA) : Stability results for set-valued optimization problems in set order functional spaces
The last ten years have seen the emergence of new challenges in financial risk management, modelling and the treatment of strong uncertainties. These modern problems generally lead to the study of optimization problems involving an objective function with values in a pre-ordered hyperspace - a convex cone. These are referred to as multivariate optimization problems, since objective functions usually have as their codomain the set of non-empty subparts of a normed vector space. Although this class of problems has been studied since the late 1990s, the work of Hamel (2005), introducing the concept of conlinear space, was the first to focus on the codomain structure of objective functions. More recently, Geoffroy and Larrouy (2022) have provided a topological framework for conlinear spaces, dedicated to the treatment of multivariate optimization problems: set order topology. From now on, we'll call “set order functional space” any space of multivariate applications whose codomain is a conlinear space equipped with set order topology.
In 2016, Gutiérrez et al. introduced the concepts of inner and outer stabilities to study multivariate optimization problems via the Kuroiwa (or setwise) approach. In this talk, we will study the inner and outer stability of multivariate optimization problems whose objective function belongs to a functional space of order over sets. We will consider perturbations on the objective function and on the admissible domain, and treat the cases of relaxed (in the sense of Geoffroy and Larrouy), Pareto and strong minimal solutions. To this end, we introduce a new variational convergence: Gamma-cone convergence, whose beautiful properties we will briefly present. To the best of our knowledge, the external stability of minimal Pareto solutions without convexity conditions on the objective function has not yet been obtained.