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Vectorization is a technique that replaces a set-valued optimization problem with a vector optimization problem. In this work, by using an extension of Gerstewitz function [1], a vectorizing function is defined to replace a given set-valued optimization problem with respect to set less order relation. Some properties of this function are studied. Also, relationships between a set-valued optimization problem and a vector optimization problem, derived via vectorization of this set-valued optimization problem, are examined. Furthermore, necessary and sufficient optimality conditions are presented without any convexity assumption.

A new relation on the subset of the space is defined via -difference in this work. This is a pre-order relation on the family of nonempty sets. Some relations between this pre-order relation and well-known order relations are investigated. Also, the solution points of a set-valued optimization problem via set and vector approaches are examined.