Weak and Proper Efficiency in Set-Valued Optimization on Real Linear Spaces

We extend the notion of cone-subconvexlikeness of set-valued maps on topological linear spaces to set-valued maps on linear spaces, (that is, general linear spaces without any particular topology), and we provide several characterizations. An alternative theorem is also established for this kind of maps. Using the notion of vector closure introduced recently by Adán and Novo, we also provide, in this framework, an adaptation of the proper efficiency in the sense of Benson for set-valued maps. The previous notion and results are then applied to obtain optimality conditions of weak efficiency and a characterization of Benson proper efficiency by means of scalarization and multipliers rules.