In this paper, we study a general optimization problem without linear structure under a reflexive and transitive relation on a nonempty set E, and characterize the existence of efficient points and the domination property for a subset of E through a generalization of the order-completeness condition introduced earlier. Afterwards, we study the abstract optimization problem by using generalized continuity concepts and establish various existence results. As an application, we extend and improve several existence results given in the literature for an optimization problem involving set-valued maps under vector and set criteria.