Conjugate Scaling Algorithm for Fenchel-Type Duality in Discrete Convex Optimization

Iwata, Satoru; Shigeno, Maiko

doi:10.1137/s1052623499352012

Cited by 34 publications

(26 citation statements)

References 20 publications

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“…Algorithms for the MSFP are given by Murota [21] and by Iwata and Shigeno [14]. Those algorithms find an optimal flow and an optimal potential, and the latter is a polynomial time algorithm in IVI and loge C, where C is a certain number satisfying C< max l y(a) l+ 2 max f(x)I.…”

Section: M-convex Submodular Flow Problemmentioning

confidence: 99%

“…Then the set P*( xo) of all the equilibrium price vectors is an La-convex polyhedron. This means in particular (a = 0 in (14)) that p,gEP*(x°) pVq, pAgEP*(x°). (15) (15) implies the existence of the smallest equilibrium price vector, and furthermore, it implies the existence of the largest equilibrium price vector if P*(x°) is bounded.…”

Section: P Q E P = (P -A1) V Q P a (Q + Al) E P (0 < Vca E R) (14)mentioning

confidence: 99%

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Application of M-Convex submodular flow problem to mathematical economics

Murota

Tamura

2003

Japan J. Indust. Appl. Math.

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This paper considers an economic model in which producers and consumers trade various indivisible commodities through a perfectly divisible commodity, money. On the basis of the recent developments in discrete mathematics (combinatorial optimization), we give an efficient algorithm to decide whether a competitive equilibrium exists or not, when cost functions of the producers are Ma-convex and utility functions of the consumers are Ma-concave and quasilinear in money, where Ma-convexity is closely related to the gross substitutes condition.

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Section: M-convex Submodular Flow Problemmentioning

confidence: 99%

Section: P Q E P = (P -A1) V Q P a (Q + Al) E P (0 < Vca E R) (14)mentioning

confidence: 99%

Application of M-Convex submodular flow problem to mathematical economics

Murota

Tamura

2003

Japan J. Indust. Appl. Math.

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“…Iwata [18] showed that L-convex functions can be minimized in polynomial time by repeatedly solving associated SFM problems obtained by scaling (also see [22]). …”

Section: Discussionmentioning

confidence: 99%

Submodular function minimization and related topics

Fujishige

2003

Optimization Methods and Software

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It had been a long-standing open problem to devise a combinatorial polynomial algorithm for minimizing submodular functions. Iwata-Fleischer-Fujishige (IFF) and Schrijver resolved the problem independently and simultaneously. The present article is basically a survey, though not comprehensive, on submodular function minimization algorithms and also describes additional results and research subjects. In this article we first give an outline of the IFF algorithm and then show two modifications of the original IFF algorithm, which seem to be promising in practical implementation. We also describe a brief history of developments in submodular function minimization. Finally, we furnish some possible future research subjects in submodular function minimization.

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“…It is known that the M -concave intersection problem for integer-valued M -concave functions can be solved in polynomial time (see [19,18]). …”

Section: A Function F : Z V → R ∪ {−∞} With Domf = ∅ Is M -Concave Ifmentioning

confidence: 99%

Applications of Discrete Convex Analysis to Mathematical Economics

Tamura

2004

Publ. Res. Inst. Math. Sci.

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Discrete convex analysis, which is a unified framework of discrete optimization, is being recognized as a basic tool for mathematical economics. This paper surveys the recent progress in applications of discrete convex analysis to mathematical economics. §1. Introduction Discrete convex analysis, proposed by Murota [25,26], is a unified framework of discrete optimization. Recently, applications of discrete convex analysis to mathematical economics have been investigated. The aim of this paper is to survey the following recent progress on this topic.The concepts of M-convex functions due to Murota [25,26] and M -convex functions due to Murota and Shioura [30], which play central roles in discrete convex analysis, are being recognized as nice discrete convex functions from the point of view of mathematical economics. For instance, for set functions, Fujishige and Yang [15] showed that M -concavity is equivalent to the gross substitutability and the single improvement property which are equivalent to each other for set functions [17] and are nice in the following sense. These properties guarantee the existence of the core of several models, e.g., a matching model proposed by Kelso and Crawford [21]. Relations among these three properties were extended to the general case by Danilov, Koshevoy and Lang [3] and by

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Conjugate Scaling Algorithm for Fenchel-Type Duality in Discrete Convex Optimization

Cited by 34 publications

References 20 publications

Application of M-Convex submodular flow problem to mathematical economics

Application of M-Convex submodular flow problem to mathematical economics

Submodular function minimization and related topics

Applications of Discrete Convex Analysis to Mathematical Economics

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