Pareto optimization in topological vector spaces

Abstract

To identify the efficient point of a set is an important problem in vector optimization. The existence of efficient points in infinite dimensional space has been studied by many authors, such as [2–10,12,15,17–19,21,23–25] etc. To study Pareto efficiency in locally convex topological vector space, in 1983, the second author Isac [14] has introduced an important notion of nuclear cone which has also other applications as, for example, in the fixed point theory, in the best approximation theory, in the study of vector optimization for multivalued functions, in the study of conically bounded sets, and in the study of nuclearity of topological vector space. About these, one can refer to Hyers–Isac–Rassias’ book [11] and references therein. In 1990, Pontini [20] defined a more general notion of pseudonuclear in general topological vector space (may not be locally convex). Pontini remarked that, in a normed vector space, a cone is pseudo-nuclear if and only if it is nuclear, but in a general topological vector space, this result is not true. In [11,13], Isac, Hyers, and Rassias discussed the existence of efficient points in locally convex topological space ordered by pointed nuclear cone and got some general results. In book [11] and paper [13], they also posed the following open problem.