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Exploitation of Applications of Discrete Convex Analysis

Research Project

Project/Area Number 12450040
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Engineering fundamentals
Research InstitutionThe University of Tokyo (2002)
Kyoto University (2000-2001)

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)

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)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥7,600,000 (Direct Cost: ¥7,600,000)
Fiscal Year 2002: ¥2,000,000 (Direct Cost: ¥2,000,000)
Fiscal Year 2001: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2000: ¥4,200,000 (Direct Cost: ¥4,200,000)
Keywordsconvex analysis / convex function / convex set / matroid / discrete optimization / mathematical programming / nonlinear programming / network flow / 離散凸解析 / 数理経済学 / 不可分財市場 / 競争均衡 / M凸関数最小化 / アルゴリズム
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.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (34 results)

All Other

All Publications (34 results)

  • [Publications] K.Murota, A.Shioura: "Extension of M-convexity and L-convexity to Polyhedral Convex Functions"Advances in Applied Mathematics. 25. 352-427 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] V.Danilov, G.Koshevoy, K.Murota: "Discrete Convexity and Equilibria in Economies with Indivisible Goods and Money"Mathematical Social Sciences. 41. 251-273 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Iwata, K.Murota: "Combinatorial Relaxation Algorithm for Mixed Polynominal Matrices"Mathematical Programming, Series A. 90.2. 353-371 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Murota, A.Tamura: "New Characterizations of M-convex Functions and Their Applications to Economic Equilibrium Models with Indivisibilities"Discrete Applied Mathematics. (to appear). (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Moriguchi, K.Murota, A.Shioura: "Scaling Algorithms for M-convex Function Minimization"IEICE Transactions on Fundamentals. E85・A. 922-929 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H.Saito, S.Matuura, T.Matsui: "A Linear Relaxation for Hub Network Design Problems"IEICE Transactions on Fundamentals. E85・A. 1000-1005 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] 室田 一雄: "離散凸解析"共立出版. 308 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Murota and A. Shioura: "Extension of M-convexity and L-convexity to Polyhedral Convex Functions"Advances in Applied Mathematics. 25. 352-427 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] V. Danilov, G. Koshevoy and K. Murota: "Discrete Convexity and Equilibria in Economies with Indivisible Goods and Money"Mathematical Social Sciences. 41. 251-273 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S. Iwata and K. Murota: "Combinatorial Relaxation Algorithm for Mixed Polynominal Matrices"Mathematical Programming, Series A. 90.2. 353-371 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Murota and A. Tamura: "New Characterizations of M-convex Functions and Their Applications to Economic Equilibrium Models with Indivisibilities"Discrete Applied Mathematics. to appear. (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S. Moriguchi, K. Murota and A. Shioura: "Scaling Algorithms for M-convex Function Minimization"IEICE Transactions on Fundamentals. E85.A. 922-929 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Saito, S. Matuura, and T. Matsui: "A Linear Relaxation for Hub Network Design Problems"IEICE Transactions on Fundamentals. E85.A. 1000-1005 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Murota: "Discrete Convex Analysis"Kyoritsu Publishing Company. (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Moriguchi, K.Murota, A.Shioura: "Scaling Algorithms for M-convex Function Minimization"IEICE Transactions on Fundamentals. E85-A. 922-929 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Moriguchi, K.Murota: "Capacity Scaling Algorithm for Scalable M-convex Submodular Flow Problems"Optimization Methods and Software. (掲載予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] A.Tamura: "On Convolution of L-convex Functions"Optimization Methods and Software. (掲載予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] H.Saito, S.Matuura, T.Matsui: "A Linear Relaxation for Hub Network Design Problems"IEICE Transactions on Fundamentals. E85-A. 1000-1005 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Matuura, T.Matsui: "New Approximation Algorithm for MAX2SAT and MAX DICUT"Journal of Operations Research Society of Japan. (掲載予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] J.Geelen, S.Iwata, K.Murota: "The Linear Delta-Matroid Parity Problem"Journal of Combinatorial Theory, Series B. (掲載予定). (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Murota: "Discrete Convex Analysis"Society for Industrial and Applied Mathematics. 380 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Murota, A.Shioura: "Relationship of M-/L-convex functions with discrete convex functions by Miller and by Favati-Tardella"Discrete Applied Mathematics. 115. 151-176 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] V.Danilov, G.Koshevoy, K.Murota: "Discrete convexity and equilibria in economies with indivisible goods and money"Mathematical Social Sciences. 41. 251-273 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Iwata, K.Murota: "Combinatorial relaxation algorithm for mixed polynomial matrices"Mathematical Programming, Series A. 90・2. 353-371 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Murota, A.Tamura: "New characterizations of M-convex functions and their applications to economic equilibrium models with indivisibilities"Discrete Applied Mathematics. (to appear). (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] S.Moriguchi, K.Murota, A.Shioura: "Scaling algorithms for M-convex function minimization"IEICE Transactions on Systems and Information. E85-D. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Y.Matsui, T.Matsui: "NP-completeness for calculating power indices of weighted majority games"Theoretical Computer Science. 263. 305-310 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 室田 一雄: "離散凸解析"共立出版. 308 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] K.Murota: "Algorithms in Discrete Convex Analysis"IEICE Transactions on Systems and Information. E83-D. 344-352 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Fujishige and K.Murota: "Notes on L-/M-convex Functions and the Separation Theorems"Mathematical Programming. 88. 129-146 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] M.Fukuda,M.Kojima,K.Murota and K.Nakata: "Exploiting Sparsity in Semidefinite Programming via Matrix Completion, I : General Framework"SIAM Journal of Optimization. 11. 647-674 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Murota and A.Tamura: "On Circuit Valuation of Matroids"Advances in Applied Mathematics. 26(掲載予定). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] S.Iwata and K.Murota: "Combinatorial Relaxation Algorithm for Mixed Polynomial Matrices"Mathematical Programming. (掲載予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] A.Tamura: "Perfect (0,±1)-Matrices and Perfect Bidirected Graphs"Theoretical Computer Science. 235. 339-356 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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