2001 Fiscal Year Final Research Report Summary
Discrete Optimization Algorithms based on Discrete Convex Analysis
| Project/Area Number |
10205212
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| Research Category |
Grant-in-Aid for Scientific Research on Priority Areas (B)
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| Allocation Type | Single-year Grants |
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| Research Institution | Kyoto University |
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Principal Investigator |
MUROTA Kazuo Kyoto University, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 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)
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)
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| Project Period (FY) |
1998 – 2000
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| Keywords | discrete optimization / convex analysis / convex function / matroid / combinatorial optimization / submodular function / convex set / base polyhedron |
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| 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
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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
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