CHARACTERIZATION OF i -BENSON PROPER EFFICIENT SOLUTIONS OF VECTOR OPTIMIZATION PROBLEMS WITH VARIABLE ORDERING STRUCTURES IN LINEAR SPACES

In this paper, using improvement-valued maps, we define two types of E-Benson proper efficient elements for subsets within a linear space under a variable ordering map l . Consequently, we delve into studying two types of E-Benson proper efficient solutions of vector optimization problems under variable ordering structures. We establish relationships among different types of E-Benson proper efficient elements. Furthermore, we demonstrate that the two types of E-Benson proper efficiency, in relation to the ordering map l, not only unify and extend certain notions of (weakly) nondominated elements but also extend some well-known notions of Benson proper efficiency under fixed ordering structures. Lastly, under suitable assumptions, we establish linear scalarization theorems for E-Benson proper efficient solutions of vector optimization problems under variable ordering structures. Several examples are also provided to illustrate the derived results.