Studies on discrete convex analysis and discrete fixed point theorems

2014 Fiscal Year Final Research Report

• Project/Area Number: 23540142
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kyushu University
• Principal Investigator: KAWASAKI Hidefumi 九州大学, 数理(科)学研究科(研究院), 教授 (90161306)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: 最適化理論 / ゲーム理論 / 不動点定理 / ナッシュ均衡 / 組合せ最適化 / 離散凸解析

Outline of Final Research Achievements

We studied continuous and discrete structure in optimization theory and game theory with fixed point theorems and convex analysis at the core. We have verified theorems of discrete convex analysis and obtained some extensions. We gave a new proof of convex extension of L-convex functions and proved the inverse of the theorem. We have extended the separation theorem of L-convex sets to three or more L-convex sets. We have studied the relationship between Brouwer's thm., Spernar's lemma, Hex thm., and so on, and presented a sufficient condition that a mapping from the vertices of equilateral subdivision of the standard simplex into itself has a discrete fixed point on each completely labeled subsimplex. We have typed 260 pages of a book titled "Continuous and discrete structure of equilibria and extrema".

Free Research Field

最適化理論