Positive Proper Efficient Points and Related Cone Results in Vector Optimization Theory

Abstract

Positive proper efficient points are defined as solutions of appropriate linear scalar optimization problems. A geometric characterization of positive proper efficient points is given as well as conditions under which the set of positive proper efficient points is dense in the set of all efficient points. It is shown that these results are applicable in the normed vector lattices $C[a,b]$, $l^p $, and $L^p $ for $1 \leqq p \leqq \infty $, and that previous related results, which required the ordering cone to have a compact or weak-compact base, are not applicable in many normed vector lattices, including $C[a,b]$, $l^p $, and $L^p $ for $1 \leqq p \leqq \infty $.