Nonlinear Scalarization Characterizations of E-Super Efficient Solution of Set-valued Vector Optimization Problem

Abstract

Abstract: In the study of vector optimization problems, the concept of (weakly) effective solution based on the definition of order cone and its properties play a very important role. The concept of superefficient solution is the unification of several existing concept of properly efficient solutions, while the E- super efficient solution of the vector optimization problem proposed based on the improvement set is an important extension of the classical concept of superefficient solution, which unifies the concepts of superefficient solution and ε-super efficient solution. Therefore, it is of great significance to study the E- super efficient solution and its related properties for the study of vector optimization problems. This paper mainly focuses on the relevant conditions of nonlinear scalarization characterization of E- super efficient solution: firstly, a sufficient and necessary condition for the nonlinear scalarization characterization about the E- super efficient solution of the set-valued vector optimization problem is obtained through the corresponding property of the seminorm p; secondly, the corresponding property of the distance function d is used to obtain another sufficient and necessary condition for the nonlinear scalarization characterization of the E- super efficient solution; then, according to the relationship between the E-optimal solution and the E-super efficient solution, a sufficient condition for the results of nonlinear scalarization characterization of the E- super efficient solution is derived by means of the distance-oriented function Δ-E which based on the improved set E under the condition of E=E+K0; finally, under the condition of 0∈clE, the distance-oriented function Δ-E is used to obtain a necessary condition for the nonlinear scalarization result of E- super efficient solution, and the corresponding sufficient and necessary result is deduced.