Directed discrete midpoint convexity

Tamura, Akihisa; Tsurumi, Kazuya

doi:10.1007/s13160-020-00416-0

Cited by 9 publications

(10 citation statements)

References 32 publications

(82 reference statements)

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“…The polyhedral description of multimodular sets, described in Theorem 2.5, has recently been obtained in [15]. Polyhedral descriptions are not known for integrally convex sets [18,Section 3.4], discrete midpoint convex sets [16], and directed discrete midpoint convex sets [30].…”

Section: Discussionmentioning

confidence: 99%

Note on the Polyhedral Description of the Minkowski Sum of Two L-convex Sets

Moriguchi¹,

Murota²

2021

Preprint

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L-convex sets are one of the most fundamental concepts in discrete convex analysis. Furthermore, the Minkowski sum of two L-convex sets, called L 2 -convex sets, are the most intriguing objects that are closely related to polymatroid intersection. This paper reveals the polyhedral description of an L 2 -convex set, together with the observation that the convex hull of an L 2 -convex set is a box-TDI polyhedron. The proofs utilize Fourier-Motzkin elimination and the obtained results admit natural graph representations. Implications of the obtained results in discrete convex analysis are also discussed.

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Section: Discussionmentioning

confidence: 99%

Note on the Polyhedral Description of the Minkowski Sum of Two L-convex Sets

Moriguchi¹,

Murota²

2021

Preprint

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“…be expressed as the sum of univariate convex, diff-convex, and sum-convex functions, i.e., if where i , ij , ij ∶ ℤ → ℝ ∪ {+∞} (i, j = 1, … , n; i ≠ j) are univariate convex functions. A 2-separable convex function is known [29] to be integrally convex. A quadratic function f (x) = x ⊤ Qx with Q satisfying (2.8) is an example of a 2-separable convex function.…”

Section: Remark 21mentioning

confidence: 99%

“…Moreover, BS-convex and UJ-convex functions [8] are integrally convex. Discrete midpoint convex functions [15] and directed discrete midpoint convex functions [29] are also integrally convex.…”

Section: Introductionmentioning

confidence: 99%

Discrete Fenchel duality for a pair of integrally convex and separable convex functions

Murota

Tamura

2022

Japan J. Indust. Appl. Math.

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Discrete Fenchel duality is one of the central issues in discrete convex analysis. The Fenchel-type min–max theorem for a pair of integer-valued M$$^{\natural }$$ ♮ -convex functions generalizes the min–max formulas for polymatroid intersection and valuated matroid intersection. In this paper we establish a Fenchel-type min–max formula for a pair of integer-valued integrally convex and separable convex functions. Integrally convex functions constitute a fundamental function class in discrete convex analysis, including both M$$^{\natural }$$ ♮ -convex functions and L$$^{\natural }$$ ♮ -convex functions, whereas separable convex functions are characterized as those functions which are both M$$^{\natural }$$ ♮ -convex and L$$^{\natural }$$ ♮ -convex. The theorem is proved by revealing a kind of box integrality of subgradients of an integer-valued integrally convex function. The proof is based on the Fourier–Motzkin elimination.

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“…Moreover, BS-convex and UJ-convex functions [7] are integrally convex. Discrete midpoint convex functions [13] and directed discrete midpoint convex functions [26] are also integrally convex.…”

Section: Introductionmentioning

confidence: 99%

Discrete Fenchel Duality for a Pair of Integrally Convex and Separable Convex Functions

Murota¹,

Tamura²

2021

Preprint

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Discrete Fenchel duality is one of the central issues in discrete convex analysis. The Fenchel-type min-max theorem for a pair of integer-valued M ♮ -convex functions generalizes the min-max formulas for polymatroid intersection and valuated matroid intersection. In this paper we establish a Fenchel-type min-max formula for a pair of integervalued integrally convex and separable convex functions. Integrally convex functions constitute a fundamental function class in discrete convex analysis, including both M ♮convex functions and L ♮ -convex functions, whereas separable convex functions are characterized as those functions which are both M ♮ -convex and L ♮ -convex. The theorem is proved by revealing a kind of box integrality of subgradients of an integer-valued integrally convex function. The proof is based on the Fourier-Motzkin elimination.

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Directed discrete midpoint convexity

Cited by 9 publications

References 32 publications

Note on the Polyhedral Description of the Minkowski Sum of Two L-convex Sets

Note on the Polyhedral Description of the Minkowski Sum of Two L-convex Sets

Discrete Fenchel duality for a pair of integrally convex and separable convex functions

Discrete Fenchel Duality for a Pair of Integrally Convex and Separable Convex Functions

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