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<p>This paper gives two different proofs to a structural theorem of decreasing minimization (lexicographic optimization) on integrally convex sets. The theorem states that the set of decreasingly minimal elements of an integrally convex set can be represented as the intersection of a unit discrete cube and a face of the convex hull of the given integrally convex set. The first proof resorts to the Fenchel-type duality theorem in discrete convex analysis and the second is more elementary using Farkas’ lemma.</p>

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Two proofs of a structural theorem of decreasing minimization on integrally convex sets

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月18日

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  • Kazuo Murota,
  • Akihisa Tamura

摘要

This paper gives two different proofs to a structural theorem of decreasing minimization (lexicographic optimization) on integrally convex sets. The theorem states that the set of decreasingly minimal elements of an integrally convex set can be represented as the intersection of a unit discrete cube and a face of the convex hull of the given integrally convex set. The first proof resorts to the Fenchel-type duality theorem in discrete convex analysis and the second is more elementary using Farkas’ lemma.