Some Characterizations of Ideal Points in Vector Optimization Problems

Abstract

Ideal points and efficient elements play an important role in the investigation of multiobjective optimization problems see, e.g., 1–9 and references therein . Recently, some relations between ideal points and efficient elements have been studied by many authors see 3, 4 . In 3 , the authors derived some sufficient conditions for the existence of ideal points in normed vector spaces. The algebra closure and vector closure were investigated in 1, 2 , which are weaker than topological closure. In this paper, we present some relations between proper ideal points and weakly, positive proper, general positive efficient points in real linear spaces. We also derive some sufficient conditions for the existence of proper ideal points and positive proper efficient points. Let X be a real linear space and A a nonempty subset of X. A is said to be a cone if λA ⊂ A for all λ > 0. A is called a convex cone if A is a cone and A A ⊂ A. A is called a pointed cone if A is a cone and A ∩ −A {0}. The algebraic interior and relative algebraic interior ofA ⊂ X are defined by, respectively,