在向量优化问题研究中, 基于序锥定义的 (弱) 有效解的概念及其性质具有十分重要的作用。超有效解的概念是对已有的几种真有效解的概念的统一, 而基于改进集而提出的向量优化问题的E-超有效解是对经典的超有效解概念的重要推广, 它统一了超有效性和ε-超有效性的概念。因此, 研究E-超有效解及其相关性质对研究向量优化问题具有十分重要的意义。本文主要对E-超有效解非线性标量化的相关条件进行研究: 首先通过半范数p的相应性质得到集值向量优化问题的E-超有效解非线性标量化的一个充要条件; 其次利用距离函数d的相关性质得到E-超有效解非线性标量化的另一个充要条件; 再次根据E-最优解与E-超有效解之间的关系, 在E=E+K₀的条件下通过基于改进集E的面向距离函数Δ_-E得出E-超有效解非线性标量化结果的一个充分条件; 最后, 在0∈clE的条件下利用面向距离函数Δ_-E得出E-超有效解非线性标量化结果的一个必要条件, 并推导出相应的充要结果。

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.