On two vectorization schemes for set-valued optimization

What the paper says

In this paper, we investigate two known solution approaches for set-valued optimization problems, both of which are based on so-called vectorization strategies. These strategies consist of deriving a parametric family of multi-objective optimization problems whose optimal solution sets approximate those of the original set-valued problem with arbitrary accuracy in a certain sense. Thus, these approaches can serve as a basis for the numerical solution of set-valued optimization problems using established solution algorithms from multi-objective optimization. We show that many properties that have already been obtained for one of the two vectorization schemes also hold for the other similarly. Thereby, it turns out that under certain assumptions there exist problem classes for both vectorization schemes in which the set-valued initial problems are even equivalent to the corresponding multi-objective replacement problems. This property is fulfilled, for example, for set-valued optimization problems with a finite feasible set, with a polytope-valued objective map, or with a convex graph. This was already known for one of the two vectorization schemes, and could now also be shown for the other scheme. • Relation between two vectorization approaches for set-valued optimization problems. • Those allow to make set-valued optimization problems numerical tractable by formulating related finite-dimensional multi-objective optimization problems. • Showing the exactness for these replacement problems for certain classes of set-valued problems and closing thereby gaps in the theory.