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<p>We present a necessary and sufficient condition for approximate proper efficiency in vector optimization problems with the ordering cone being a nonnegative orthant. Although the criterion can be established by Benson’s approach (J. Math. Anal. Appl. <b>71</b>, 232–241, <CitationRef CitationID="CR1">1979</CitationRef>), detailed proofs are given for the first time here. The criterion is a strong motivation to introduce the concept of <i>e</i>-properly efficient solution, where <i>e</i> is any nonzero vector taken from the closed pointed ordering cone. For an arbitrary linear vector optimization problem, we show that either the <i>e</i>-properly efficient solution set is empty or it coincides with the <i>e</i>-efficient solution set. Several illustrative examples are provided.</p>

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Approximate Proper Efficiency in Vector Optimization via Benson’s Approach

  • 期刊论文

  • 发表日期：

    2026年4月25日

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  • Nguyen Thi Thu Huong

摘要

We present a necessary and sufficient condition for approximate proper efficiency in vector optimization problems with the ordering cone being a nonnegative orthant. Although the criterion can be established by Benson’s approach (J. Math. Anal. Appl. 71, 232–241, 1979), detailed proofs are given for the first time here. The criterion is a strong motivation to introduce the concept of e-properly efficient solution, where e is any nonzero vector taken from the closed pointed 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. Several illustrative examples are provided.