Scalarization in set optimization with solid and nonsolid ordering cones

This paper focuses on characterizations via scalarization of several kinds of minimal solutions of set-valued optimization problems, where the objective values are compared through relations between sets (set optimization). For this aim we follow an axiomatic approach based on general order representation and order preservation properties, which works in any abstract set ordered by a quasi order (i.e., reflexive and transitive) relation. Then, following this approach, we study a recent Gerstewitz scalarization mapping for set-valued optimization problems with K-proper sets and a solid ordering cone K. In particular we show a dual minimax reformulation of this scalarization. Moreover, in the setting of normed spaces ordered by non necessarily solid ordering cones, we introduce a new scalarization functional based on the so-called oriented distance. Using these scalarization mappings, we obtain necessary and sufficient optimality conditions in set optimization. Finally, whenever the ordering cone is solid, by considering suitable generalized Chebyshev norms with appropriate parameters, we show that the three scalarizations studied in the present work are coincident.