Cross-Sectional Research of Discrete Convex Analysis

2019 Fiscal Year Final Research Report

• Project/Area Number: 26280004
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Mathematical informatics
• Research Institution: Tokyo Metropolitan University (2016-2019); The University of Tokyo (2014-2015)
• Principal Investigator: MUROTA Kazuo 首都大学東京, 経営学研究科, 教授 (50134466)
• Project Period (FY): 2014-04-01 – 2020-03-31
• Keywords: 離散凸解析 / 最適化理論 / 数理工学 / 情報基礎 / アルゴリズム / 経済理論 / ゲーム理論 / オペレーションズ・リサーチ

Outline of Final Research Achievements

Discrete Convex Analysis is a theory of optimization that transversally connects the continuous optimization and discrete optimization. The theory aims at establishing a general methodology applicable in various disciplines of engineering and social science by extracting and reformulating common mathematical structures found in various problems. In this research project we have obtained results on discrete DC (difference of convex) programming, integrally convex functions and discrete midpoint convex functions, axiomatizations and characterizations of M-convex functions, operations on discrete convex functions, etc. We have also explored applications of discrete convex analysis to auction theory in economics and game theory, inventory theory in operations research, valued constrained satisfaction problem in computer science.

Free Research Field

数理工学

Academic Significance and Societal Importance of the Research Achievements

離散凸解析は，最適化において「連続と離散を繋ぐパラダイム」であり，様々な分野で別々に考察されてきた数学的な構造を，分野を越えて理解して，相互に利用するための枠組みである．離散凸解析の理論やアルゴリズムが一般的な形で整理されることによって，コンピュータ科学，オペレーションズ・リサーチ，経済学，ゲーム理論，数学などの様々な分野での共通の言葉やアプローチが生まれ，学問諸分野の交流が可能となる．さらには，その共通の知識に基づいて，様々な応用が繋がり発展していくことが期待される．