Approximate proper solutions in vector optimization with variable ordering structure

In this paper, we study approximate proper efficient (nondominated and minimal) solutions of vector optimization problems with variable ordering structures (VOSs). In vector optimization with VOS, the partial ordering cone depends on the elements of the image set. Approximate proper efficient/nondominated/ minimal solutions are defined in different senses (Henig, Benson, and Borwein) for problems with VOSs from new standpoints. The relationships among the introduced notions are studied, and some scalarization approaches are developed to characterize these solutions. These scalarization results based on new functionals defined by elements from the dual cones are given. Moreover, some existing results are addressed.