Approximate solutions of vector optimization problems via improvement sets in real linear spaces
The authors consider the following vector optimization problem: \[ f(x)\longrightarrow \min,\text{ subject to } x \in S, \] where $X$ is an arbitrary decision space, $Y$ is a real linear space ordered by the relation \[ y_1 \leq_{K} y_2 \Longleftrightarrow y_2 -y_1 \in K, \] where $K$ is a convex cone. The notion of improvement is introduced as follows: A nonempty set $E \subset Y$ is said to be an improvement set with respect to $K$ if $E \neq \emptyset$ and $E + K = E$. A point $x^0 \in S$ is said to be $E$-optimal point of the considered vector optimization problem if \[ f(S) - f(x^0) \cap (-E) = \emptyset. \] The authors introduce several modifications of $E$-optimality and study relations between them. The $E$-optimality and its modifications are studied on a real linear space without considering any topology just by using algebraic structures.
scientific article; zbMATH DE number 6873202
scientific article