Study on Nonlinear Scalarization Methods for Set-Valued Maps and its Applications into Mathematical Programming Problems and Statistical Science.

在本研究中，我们对集值映射的向量优化和非线性标量化方法进行了分析和广泛的研究，1.在第四届国际非线性分析和凸分析会议上，我们发表了六篇关于非线性标量化函数的基础研究及其在博弈论中的应用研究的论文（NACA 2005，冲绳）。论文发表于2006年《中国科学院学报》。它们涉及（i）向量值函数的Caristi不动点定理，（ii）利用Fan-KKM引理得到的向量值函数锥鞍点的存在性结果，（iii）集值向量优化问题有效解的最优性条件。2.作为数学规划的一个应用，我们与山田副教授和谷野教授（大坂 ...更多信息 大学）联合。Tanino教授基于仿射尺度方法得到了全局优化问题的混沌发生器，并发表了论文; 3.作为统计科学中多维数据的应用，Isogai教授考虑了任意和可能非光滑密度的假设问题; 4.在国外，我们在第8届广义凸性/单调性国际研讨会上发表了我们的结果（瓦雷塞，意大利），第三届中日电子商务优化会议（SJOM 2005）（新加坡）。我们在中原大学、国立中山大学、国立清华大学、台湾真理大学、国立成功大学、台湾昌化教育大学做了特邀报告。5.在日本，我们在RIMS Symposium on Nonlinear Analysis and Convex Analysis上发表了我们的成果（京都大学），数学决策及其应用发展研讨会（长崎大学经济系）、数学模型的决定及其应用研讨会（高知大学理学系）。少

In this research, we have studied several kinds of themes on vector optimization and nonlinear scalarization methods for set-valued maps analytically and broadly, and then we have applied to multi-dimensional data in a statistical science.1.We presented six papers for fundamental researches on nonlinear scalarizing functions and its applied researches into game theory at the Fourth International Conferences on Nonlinear Analysis and Convex Analysis (NACA2005,Okinawa). The papers have been published in the (refereed) proceedings,2006. They are related to (i)Caristi's fixed point theorems for vector-valued functions, (ii)existence results of cone saddle-points for vector-valued functions by using the Fan-KKM lemma, (iii)optimality conditions for efficient solutions in set-valued vector optimization problems.2.As an application in mathematical programming, we researched outer approximation methods for global optimization problems with Associate Professor Yamada and Professor Tanino (Osaka … More University) jointly. Professor Tanino obtained chaos generator based on the affine scaling method for global optimization problems, and its paper was published.3.As an application into multi-dimensional data in a statistical science, Professor Isogai considered presumption problems of arbitrary and possibly non-smooth densities.4.In abroad, we presented our results in the 8-th International Symposium on Generalized Convexity/Monotonicity (Varese, Italy), the 3rd Sino-Japanese Optimization Meeting in the e-Business Era (SJOM 2005)(Singapore). We delivered invited talks at Chung Yuan Christian University, National Sun Yat-sen University, National Tsing-Hua University, Aletheia University, National Cheng Kung University, Changhua University of Education in Taiwan.5.In Japan, we presented our results in RIMS Symposium on Nonlinear Analysis and Convex Analysis (Kyoto University), the symposium on developments of mathematical decision and its application (Department of Economics at Nagasaki University), the symposium on decision to mathematical model and its application (Department of Science at Kochi University). Less