Contractibility of efficient point sets in normed spaces

Abstract

L E a topological vector space over reals, C a convex cone and X a set in E. C specifies a partial order in E as follows: x, y ∈ E, x ≥ C y if x − y ∈ C. With this order one can define the set of efficient points of X with respect to C as the set Min(X | C) = {x ∈ X: if x ≥ C y for some y ∈ X, then y ≥ C x}. Whenever the relative interior of C, riC, is nonempty, the cone {0} U riC gives another order defining the set of weakly efficient points of X with respect to C, denoted W Min(X | C). One of the most important problems of vector optimization is to investigate topological properties of the sets Min(X | C) and W Min(X | C). Our aim is to develop a new method which makes it possible to prove most important properties of efficient point sets such as contractibility and connectedness in general normed spaces