Set Approach for Set Optimization with Variable Ordering Structures Part II: Scalarization Approaches

Computing minimal elements of finite families of sets w.r.t. preorder relations in set optimization

“…Although we have implemented them only for the preorder relations ≤ C , ≤ u C and ≤ s C and a particular type of sets, for which we succeeded to construct strongly increasing sorting functions, it would be of both theoretical and practical interest for further research to consider other (preorder) relations and different classes of sets, along with appropriate strongly increasing sorting functions. In particular, since a variant of the Jahn-Graef-Younes method for vector optimization problems has been already developed for variable ordering structures by Eichfelder [3], it is natural to study whether this method could be generalized for set optimization problems in the framework considered by Eichfelder and Pilecka [1,2]. Other interesting research topics would be to involve a pre-sorting procedure in the numerical methods proposed by Köbis and Le [14] for computing strict, strong and ideal minimal elements of a finite family of sets, as well as in the numerical methods for computing approximations of minimal elements, following the works mentioned in the Introduction.…”

Unifying local–global type properties in vector optimization

“…In particular, our Theorems 9, 10 and 11 are of special interest for unified vector optimization, since they encompass the "local min -global min" type properties (when K 1 = K 2 ), as well as the "local max -global min" type properties (when K 1 = −K 2 ). An interesting topic for further research would be to establish similar local-global extremality properties for set-valued functions with respect to variable ordering structures (see, e.g., Durea, Strugariu and Tammer [9], Eichfelder and Pilecka [10]- [11], and Köbis [22]).…”

A first bibliography on set and vector optimization problems with respect to variable domination structures

“…D. On the other hand, given a nondominated element y of A w.r.t. D, a domination set D(y) is a set associated with another element y. Characterizations and important properties of nondominated-like elements and nondominated elements can be found in [25,24,30,31,32,33,34,37,38,39,40,41,42,99,100].…”