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Discrete Optimization Algorithms based on Discrete Convex Analysis

Research Project

Project/Area Number 10205212
Research Category

Grant-in-Aid for Scientific Research on Priority Areas (B)

Allocation TypeSingle-year Grants
Research InstitutionKyoto University

Principal Investigator

MUROTA Kazuo  Kyoto University, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 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)
FURIHATA Daisuke  Kyoto University, Research Institute for Mathematical Sciences, Research Associate, 数理解析研究所, 助手 (80242014)
SHIOURA Akiyoshi  Sophia University, Faculty of Science and Technology, Assistant, 理工学部, 助手 (10296882)
FUJIE Tetsuya  Kobe University of Commerce, Dept. of Management Science, Research Assistant, 管理科学科, 助手 (40305678)
Project Period (FY) 1998 – 2000
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥5,800,000 (Direct Cost: ¥5,800,000)
Fiscal Year 2000: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1999: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1998: ¥4,000,000 (Direct Cost: ¥4,000,000)
Keywordsdiscrete optimization / convex analysis / convex function / matroid / combinatorial optimization / submodular function / convex set / base polyhedron / アルゴリズム / 離散凸関数 / M凸関数 / スケーリング技法 / M凸劣モジュラ流問題 / 安定集合問題 / 付値マトロイド / 非線形計画
Research Abstract

In the area of nonlinear programming, the theory of convex analysis provides a unified framework for well-solved problems. In the area of discrete optimization, on the other hand, we do not have such a unified framework. Matroidal structure, however, is known to be a well-behaved structure in discrete optimization. The theory of "discrete convex analysis" is proposed by the head investigator with a view to capturing the continuous optimization and the discrete optimization from a common viewpoint. Discrete convex analysis is a theoretical framework connecting the theory of convex analysis and the theory of matroids.
It is often the case that discrete optimization problems appearing in the real world do not have nice combinatorial structure such as matroids. Therefore, we need to use enumeration-type algorithms such as the branch-and-bound method or approximation algorithms such as meta heuristics. In either approach, it is essential to extract tractable part from the discrete structure … More of the problems to be solved. For example, it is often effective to extract network-like structure as subproblems when we solve general discrete optimization problems.
The fundamental idea of our research is to extract certain structure which can be dealt with discrete convex analysis as subproblems when we solve general discrete optimization problems. We summarize the main results as follows :
・The concepts of M-convex and L-convex functions play a central role in the framework of discrete convex analysis. We showed various properties of these functions.
・We proposed efficient algorithms for the minimization of M-convex functions.
・We extended the concepts of M-convex and L-convex functions defined over the integer lattice to functions over the real space.
・We generalized the concepts of M-convex and L-convex functions to quasi M-convex/L-convex functions.
・We obtained some results on the application of discrete convex analysis to economic equilibrium such as the equivalence of gross substitutes property and M-convexity. Less

Report

(4 results)
  • 2001 Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (35 results)

All Other

All Publications (35 results)

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

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Murota, K., Tamura, A.: "New Characterizations of M-convex Functions and Their Applications to Economic Equilibrium Models"Discrete Applied Mathematics. (to appear). (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Furihata, D.: "A Stable and Conservative Finite Difference Scheme for the Cahn-Hilliard Equation"Numerische Mathematik. 87. 675-699 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shinano, Y., Fujie, T.: "Parallel Branch-and-Bound Algorithms on a PC Cluster using PUBB"Proc.of World Multiconference on Systemics,Cybernetics and Informatics. 3. 328-333 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] McCormick, S.T., Shioura, A.: "Minimum Ratio Canceling is Oracle Polynomial for Linear Programming,but Not Strongly Polynomial,Even for Networks"Operations Research Letters. 27. 199-207 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 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
      2001 Final Research Report Summary
  • [Publications] Murota, K., Shioura, A.: "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
      2001 Final Research Report Summary
  • [Publications] Murota, K., Shioura, A.: "Relationship of M-/L-convex Functions with Discrete Convex Functions by Miller and by Favati-Tardella"Discrete Applied Mathematics. 115. 151-176 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Murota, K., Tamura, A.: "New Characterizations of M-convex Functions and Their Applications to Economic Equilibrium Models"Discrete Applied Mathematics. (to appear). (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Furihata, D.: "A Stable and Conservative Finite Difference Scheme for the Cahn-Hilliard Equation"Numerische Mathematik. 87. 675-699 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Shinano. Y., Fujie, T.: "Parallel Branch-and-Bound Algorithms on a PC Cluster using PUBB"Proc. of World Multiconference on Systemics, Cybernetics and Informatics. 3. 328-333 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] McCormick, S. T., Shioura, A.: "Minimum Ratio Canceling is Oracle Polynomial for Linear Programming, but Not Strongly Polynomial, Even for Networks"Operations Research Letters. 27. 199-207 (2000)

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

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Murota,K.: "Algorithms in Discrete Convex Analysis"IEICE Transactions on Systems and Information. E83-D. 344-352 (2000)

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

    • Related Report
      2000 Annual Research Report
  • [Publications] Nakamura,D.,Tamura,A.: "A Linear Time Algorithm for the Generalized Stable Set Problem on Triangulated Bidirected Graphs"Journal of the Operations Research Society of Japan. 43. 162-175 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Nakamura,D.,Tamura,A.: "A Revision of Minty's Algorithm for Finding a Maximum Weight Stable Set of Claw-Free Graph"Journal of the Operations Research Society of Japan. (掲載予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] McCormick,S.T.,Shioura,A.: "Minimum Ratio Canceling is Oracle Polynomial for Linear Programming, but Not Strongly Polynomial, Even for Networks"Operations Research Letters. 27. 199-207 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Furihata,D.: "A Stable and Conservative Finite Difference Scheme for the Cahn-Hilliard Equation"Numerische Mathematik. (掲載予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] Tamura,A.: "一般化安定集合問題(離散構造とアルゴリズムVII、第2章)"近代科学社. 46 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] K. Murota: "Submodular Flow Problem with a Nonseparable Cost Function"Combinatorica. 19. 87-109 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K. Murota and A. Shioura: "M-Convex Function on Generalized Polymatroid"Mathematics of Operations Research. 24・1. 95-105 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] S. T. McCormick and A. Shioura: "Minimum Ratio Canceling is Oracle Polynomial for Linear Programming, but Not Strongly Polynomial, Even for Networks"Operations Research Letters. (掲載予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] A. Shioura: "Level Set Characterization of M-convex Functions"IEICE Transactions. (掲載予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] A. Tamura: "Perfect (0, ±1)-Matrices and Perfect Bidirected Graphs"Theoretical Computer Science. (掲載予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] D. Furihata: "Finite Difference Schemes for (∂u)/(∂t) = (∂/(∂x))^α(δG)/(δu) That Inherit Energy Conservation or Dissipation Property"Journal of Computational Physics. 156. 181-205 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] K. Murota: "Matrices and Matroids for Systems Analysis (Algorithms and Combinatorics 20)"Springer-Verlag. 483 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] K.Murota: "Fenchel-type Duality for Matroid Valuations" Mathematical Programming. 82. 357-375 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Murota: "Discrete Convex Analysis" Mathematical Programming. 83. 313-371 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Murota: "On the Degree of Mixed Polynomial Matrices" SIAM Journal on Matrix Analysis and Applications. 20. 196-227 (1999)

    • Related Report
      1998 Annual Research Report
  • [Publications] K.Murota,A.Shioura: "M.Convex Function on Generalized Polymatroid" Mathematics of Operations Research,to appear.

    • Related Report
      1998 Annual Research Report
  • [Publications] A.Shioura: "A Constructive Proof for the Induction of M-Convex Functions through Networks" Discrete Applied Mathematics. 82. 271-278 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] A.Shioura: "Minimization of an M-convex Function" Discrete Applied Mathematics. 84. 215-220 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] 室田一雄、他: "離散構造とアルゴリズム、V" 近代科学社, 249 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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