Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two Proofs of a Structural Theorem of Decreasing Minimization on Integrally Convex Sets

Kazuo Murota, Akihisa Tamura

TL;DR

The paper addresses the structure of decreasing-minimization (lexicographic optimization) for integrally convex sets, proving that the dec-min element set equals the intersection of a convex-hull face with a unit-size integer box. It provides two proofs of this structural theorem: (i) a Fenchel-type duality argument in discrete convex analysis and (ii) an elementary approach based on Farkas' lemma that avoids duality. The results generalize known facts for M-convex and M2-convex sets by situating decmin(S) within a convex-geometric description, namely decmin(S) = F cap [z, z']_Z with 0 ≤ z' − z ≤ 1. Together, the approaches yield a compact, convex-analytic understanding of dec-min elements with constructive implications for identifying the supporting face F and the bounding box.

Abstract

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.

Two Proofs of a Structural Theorem of Decreasing Minimization on Integrally Convex Sets

TL;DR

The paper addresses the structure of decreasing-minimization (lexicographic optimization) for integrally convex sets, proving that the dec-min element set equals the intersection of a convex-hull face with a unit-size integer box. It provides two proofs of this structural theorem: (i) a Fenchel-type duality argument in discrete convex analysis and (ii) an elementary approach based on Farkas' lemma that avoids duality. The results generalize known facts for M-convex and M2-convex sets by situating decmin(S) within a convex-geometric description, namely decmin(S) = F cap [z, z']_Z with 0 ≤ z' − z ≤ 1. Together, the approaches yield a compact, convex-analytic understanding of dec-min elements with constructive implications for identifying the supporting face F and the bounding box.

Abstract

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.
Paper Structure (8 sections, 10 theorems, 73 equations, 2 figures)

This paper contains 8 sections, 10 theorems, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Integrally convex sets
  4. Convex characterization of dec-minimality
  5. Proof Based on Fenchel-type Duality
  6. Alternative Elementary Proof
  7. Bounding by a unit box
  8. Determining a face

Key Result

Theorem 1.1

An M-convex set $S$ has a dec-min element. The set of dec-min elements of $S$ can be represented as with an integer vector $z$ and a matroid basis family $\mathcal{B}$. In particular, ${\rm decmin}(S)$ is an M-convex set.

Figures (2)

  • Figure 1: Partition $\{ V_{j} \}$ of $N$ ($x$: = =, $y$: ---)
  • Figure 2: Definition of boxes $B\sp{\circ}$, $B$, and $B'$; $\bullet \in {\rm decmin}(S)$, $\circ \in S \setminus {\rm decmin}(S)$

Theorems & Definitions (43)

  • Theorem 1.1: FM22partA
  • Theorem 1.2: FM22fairsbmflow, Mdcamarz24
  • Theorem 1.3: Mdcamarz24
  • Example 1.1
  • Theorem 2.1: FM19partII, Mdcamarz24
  • Theorem 3.1: MT23ICsurv
  • Proposition 3.2
  • proof
  • Remark 3.1
  • Example 3.1
  • ...and 33 more