On the Scalarization of Set-Valued Optimization Problems with Respect to Total Ordering Cones

Abstract

A construction method of total ordering cone on n dimensional Euclidean space was given, it was shown that any total ordering cone is isomorphic to the lexicographic cone, also, existence of a total ordering cone that contain given cone with a compact base was shown and by using this cone, a solving method of vector and set-valued optimization problems was given recently by Kucuk et.al. In this work, it is shown that the minimal element for the set-valued optimization problem with respect to the total ordering cone is also minimal element for the corresponding optimization problem. In addition, we give examples that show the absence of the relationships between the continuity of a set valued map and K-minimal element of this map.