2002 Fiscal Year Final Research Report Summary
Exploitation of Applications of Discrete Convex Analysis
| Project/Area Number |
12450040
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Engineering fundamentals
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| Research Institution | The University of Tokyo (2002)
Kyoto University (2000-2001) |
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Principal Investigator |
MUROTA Kazuo Graduate School of Information Sciences and Technology, Professor, 大学院・情報理工学系研究科, 教授 (50134466)
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| Co-Investigator(Kenkyū-buntansha) |
TAMURA Akihisa Kyoto University, Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (50217189)
MATUURA Shiro Graduate School of Information Sciences and Technology, Research Associate, 大学院・情報理工学系研究科, 助手 (00332619)
MATSUI Tomomi Graduate School of Information Sciences and Technology, Associate Professor, 大学院・情報理工学系研究科, 助教授 (30270888)
FURIHATA Daisuke Osaka University, Cybermedia Center, Associate Professor, サイバーメディアセンター, 助教授 (80242014)
SHIOURA Akiyoshi Tohoku University, Graduate School of Information Sciences, Associate Professor, 大学院・情報科学研究科, 助教授 (10296882)
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| Project Period (FY) |
2000 – 2002
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| Keywords | convex analysis / convex function / convex set / matroid / discrete optimization / mathematical programming / nonlinear programming / network flow |
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| Research Abstract |
The main aim of this research project is to exploit applications of the theory of discrete convex analysis proposed by the head investigator in the last decade. In this research project, we obtained various results described below. All of the results are presented at refereed international conferences and/ or accepted for publications in international scientific journals.
(1) In the area of mathematical economics we obtained three results as applications of discrete convex analysis. First of all, we developed a theory for economic equilibrium models with indivisible goods ; in particular, we investigated sufficient conditions for the existence of competitive equilibrium in an exchange economy with indivisible goods. Secondly, we provide new characterizations of M-convex functions which play a central role in discrete convex analysis. These characterizations are based on well-known concepts in mathematical economics such as the gross substitutes condition and the single improvement condition. Thirdly, we developed an algorithm for computing a competitive equilibrium in an exchange economy with indivisible goods. This algorithm is based on M-convex submodular flow problem which is an optimization problem in discrete convex analysis.
(2) We showed a proximity theorem for M-convex function minimization and developed fast algorithms.
(3) We discussed combinatorial structure in engineering system from the viewpoint of discrete convex analysis. In particular, we investigated the delta matroid parity problem with the aim of determining the solvability of electrical circuits, and developed an efficient algorithm.
(4) We verified applicability of the algorithms in discrete convex analysis to several problems in operations research. Based on this result, we developed efficient algorithms for the network design problem and the max cut problem.
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