On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces
In vector optimization, topological properties of the set of efficient and weak efficient points are of interest. We study the connectedness of the set \(E^ w\) of all weakly efficient points of a subset Z of a locally convex space X with respect to a continuous mapping p: \(X\to Y\), Y locally convex and partially ordered by a closed, convex cone with nonempty interior. Under the general assumptions that Z is convex and closed and that p is a pointwise quasiconvex mapping (i.e., a generalizd quasiconvex concept), the set \(E^ w\) is connected, if the lower level- sets of p are compact. Furthermore, we show some connectedness results on the efficient points and the efficient and weakly efficient outcomes. The considerations of this paper extend the previous results of \textit{E. U. Choo} and \textit{D. R. Atkis} [ibid. 36, 203-220 (1982; Zbl 0452.90076); Manage Sci. 29, 250-255 (1983; Zbl 0519.90082)] and \textit{A. R. Warburton} [J. Optimization Theory Appl. 40, 537-557 (1983; Zbl 0496.90073)]. Moreover, some examples in vector approximation are given.
scientific article; zbMATH DE number 4104738
scientific article