The Structure of the Efficient Frontier of Finite-Dimensional Completely-Shaded Sets1

The aim of this paper is to study the contractibility of the efficient frontier of closed free-disposal sets with respect to general ordering cones in the Euclidean space. In fact, it is shown that, under certain suitable continuity assumptions, the efficient frontier of a completely-shaded set is a strong deformation retract of its weakly-efficient frontier, the last one being homeomorphic to a hyperplane. In particular, for the standard positive cone, we can apply our result to multicriteria optimization to get the contractibility of the efficient outcome set.