This work concerns with a vector optimization problem with set-valued mappings. In solving this problem, weakly efficient solutions with respect to the so-called vector criterion are considered. These solutions are defined via the notion of quasi interior, in order to provide useful results to certain problems where the topological and algebraic interior of the ordering cone is empty. Moreover, the domination set that defines the domination structure of the problem is assumed to be free disposal with respect to a convex cone. In this setting, optimality conditions via linear scalarization results and Lagrangian multiplier theorems are stated. Some of them improve several recent ones of the literature, since they are obtained under weaker assumptions
