Two Generalizations of a Theorem of Arrow, Barankin, and Blackwell

On Connectedness of Efficient Solution Sets for a Star Cone-Quasiconvex Vector Optimization Problem

“…For example, the nonnegative orthants in l p and L p (Ω), 1 ≤ p < ∞ neither have a nonempty interior nor a weakly compact base. In this paper, we shall prove the same results under the assumption that the cone is a D-cone introduced in [6]. A D-cone includes any closed convex pointed cones in a normed space which admits strictly positive continuous linear functionals, and in particular, the nonnegative orthants in C[a, b], l p , and L p (Ω) for 1 ≤ p ≤ ∞.…”

“…Moreover, if K is a nonempty convex cone of a locally convex space Y , then K +i = ∅ if and only if K has a base (see [11]). We mention specially that the nonnegative orthants in many normed vector lattice including C[a, b], l p , and L p (Ω) for 1 ≤ p ≤ ∞ admit a base, but not a bounded base except in l 1 , l ∞ and L 1 (Ω), L ∞ (Ω) (see [6]). …”

“…Some sufficient conditions for a cone to be a D-cone are given in the following Lemma (see Lemma 2.5 in [6]). …”

“…A typical example of a dual system is the case when Y is a locally convex space and Y * is its topological dual. We recall the notion of a D-cone introduced in [6] which plays an important role in the proof of density theorem. Definition 2.1 Let (Y, Y * ) be a dual system (in particular, Y is a locally convex space).…”

Density theorems for ideal points in vector optimization

“…In particular, many extensions of the above result of Arrow, Barankin and Blackwell have been given by several authors (see, for instance, Ferro, 1993;Gallagher and Saleh, 1993;Chen, 1995).…”