On Supremum of a Set in a Multi-dimensional Space

Several kinds of supremum of a set in a multi-dimensional Euclidean space have been defined (Zowe [S], Gross [2]. Nieuwenhuis [S], Brumelle [ 11, Ponstein [6], Kawasaki [3], and so on). In this paper, we reconsider the definition of supremum as an extension of the ordinary supremum in the uni-dimensional space. The most appropriate definition should satisfy some extensions of the desirable properties of the ordinary supremum and be useful for the analysis in vector optimization. As a conclusion, the definition based on the weak efficiency seems to be the most appropriate from the above mathematical viewpoint. We will give an exact definition of supremum of a set in the extended multi-dimensional Euclidean space which has the two additional elements + co. Our definition is shown to be almost equivalent to that by Kawasaki [3]. Some properties of our supremum will be investigated. It will be useful for developing conjugate duality theory in vector optimization (Kawasaki [4] and Sawaragi et al. [TJ).