Project/Area Number  10205212 
Research Category 
GrantinAid for Scientific Research on Priority Areas (B).

Research Institution  Kyoto University 
Principal Investigator 
MUROTA Kazuo Kyoto University, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (50134466)

CoInvestigator(Kenkyūbuntansha) 
SHIOURA Akiyoshi Sophia University, Faculty of Science and Technology, Assistant, 理工学部, 助手 (10296882)
FURIHATA Daisuke Kyoto University, Research Institute for Mathematical Sciences, Research Associate, 数理解析研究所, 助手 (80242014)
TAMURA Akihisa Kyoto University, Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (50217189)
FUJIE Tetsuya Kobe University of Commerce, Dept. of Management Science, Research Assistant, 管理科学科, 助手 (40305678)

Project Fiscal Year 
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)

Keywords  discrete optimization / convex analysis / convex function / matroid / combinatorial optimization / submodular function / convex set / base polyhedron / 離散最適化 / 凸解析 / 凸関数 / マトロイド / 組合せ最適化 / 劣モジュラ関数 / 凸集合 / 基多面体 / 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 wellsolved 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 wellbehaved 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 enumerationtype algorithms such as the branchandbound method or approximation algorithms such as meta heuristics. In either approach, it is essential to extract tractable part from the discrete structure
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of the problems to be solved. For example, it is often effective to extract networklike 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 Mconvex and Lconvex 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 Mconvex functions. ・We extended the concepts of Mconvex and Lconvex functions defined over the integer lattice to functions over the real space. ・We generalized the concepts of Mconvex and Lconvex functions to quasi Mconvex/Lconvex functions. ・We obtained some results on the application of discrete convex analysis to economic equilibrium such as the equivalence of gross substitutes property and Mconvexity. Less
