The first part of this paper is devoted to consideration on the definition of "supremum" in a multi-dimensional Euclidean space. A desirable definition is looked for among several possible alternatives. In the second part conjugate duality in multiobjective optimization is developed. Supremum is defined in the extended multi-dimensional Euclidean space on the basis of consideration in the first part. Some useful concepts such as conjugate maps and subgradients are introduced for vector-valued set-valued maps. Finally a strong duality result for a multiobjective optimization problem is proved under a regularity condition
