基于Benson方法的向量优化中的近似真有效性
Approximate Proper Efficiency in Vector Optimization via Benson's Approach
AI总结：
本文基于Benson方法提出两个检验向量优化近似真有效性的准则，首次给出详细证明，并引入$e$-真有效解概念，证明对任意线性向量优化问题，$e$-真有效解集要么为空，要么与$e$-有效解集重合。
摘要：
我们提出了两个准则，用于检验向量优化问题中排序锥为非负正交锥时的近似真有效性。尽管这些准则可以通过Benson的方法建立[H.P. Benson, \ extit{An improved definition of proper efficiency for vector maximization with respect to cones}, J. Math. Anal. Appl. \ extbf{71} (1979), 232--241]，但本文首次给出了详细的证明。这两个准则为引入$e$-真有效解的概念提供了强有力的动机，其中$e$是从闭尖凸排序锥中取出的任意非零向量。对于任意的线性向量优化问题，我们证明$e$-真有效解集要么为空，要么与$e$-有效解集重合。这一新结果在文献中尚无类似结论。
英文摘要：
We present two criteria for checking approximate proper efficiency in vector optimization problems with the ordering cone being a nonnegative orthant. Although the criteria can be established by Benson's approach [H.P. Benson, \textit{An improved definition of proper efficiency for vector maximization with respect to cones}, J. Math. Anal. Appl. \textbf{71} (1979), 232--241], detailed proofs are given for the first time here. The two criteria are strong motivations to introduce the concept of $e$-properly efficient solution, where $e$ is any nonzero vector taken from the closed pointed convex ordering cone. For an arbitrary linear vector optimization problem, we show that either the $e$-properly efficient solution set is empty or it coincides with the $e$-efficient solution set. This new result has no analogue in the literature.
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