A survey of fundamental operations on discrete convex functions of various kinds

Murota, Kazuo

doi:10.1080/10556788.2019.1692345

Cited by 20 publications

(18 citation statements)

References 47 publications

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“…The statements made in Remark 2.1 can be adapted to L ♮ -convexity. The reader is referred to [18,Section 5.5] and [22,Proposition 2.3] for characterizations of L ♮ -convex sets.…”

Section: L-convexitymentioning

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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“…The statements made in Remark 2.1 can be adapted to L ♮ -convexity. The reader is referred to [18,Section 5.5] and [22,Proposition 2.3] for characterizations of L ♮ -convex sets.…”

Section: L-convexitymentioning

confidence: 99%

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

Moriguchi¹,

Murota²

2021

Preprint

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“…• The addition of (4.18) and (4. 19) results in an obvious inequality α ℓ ≤ β ℓ . According to the value of a k,ℓ+1 ∈ {+1, −1, 0}, this inequality is contained in (4.15), (4.16), or (4.17) for ℓ + 1, and hence is contained in IQ(ℓ + 1).…”

Section: Fourier-motzkin Elimination For ∂ F (X) ∩ Bmentioning

confidence: 99%

“…A proximity theorem for integrally convex functions is established in [12] together with a proximity-scaling algorithm for minimization. Fundamental operations for integrally convex functions such as projection and convolution are investigated in [11,19,20]. Integer-valued integrally convex functions enjoy integral biconjugacy [22].…”

Section: Introductionmentioning

confidence: 99%

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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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“…A proximity theorem for integrally convex functions is established in [14] together with a proximity-scaling algorithm for minimization. Fundamental operations for integrally convex functions such as projection and convolution are investigated in [13,21,22]. Integer-valued integrally convex functions enjoy integral biconjugacy [24].…”

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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A survey of fundamental operations on discrete convex functions of various kinds

Cited by 20 publications

References 47 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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