Analytical Study on Vector Optimization and its Applications into Mathematical Programming Problems

本论文对向量优化的几个方面进行了分析和广泛的研究，并对集值映射的鞍点存在性、向量平衡问题、凸性和不连续性等问题进行了系统的研究和探讨，得到了一些结果。此外，我们还研究了多目标选址问题的有效解。1.我们在2001年8月和2003年8月分别在日本广崎大学和东京工业大学举行的第二届和第三届非线性分析和凸分析国际会议（NACA 2001，NACA 2003）上，介绍了我们关于有效解的特征和锥鞍点的存在性的结果。2003年11月，在NACA 2003和非线性分析与凸分析研讨会上发表了关于向量值函数锥鞍点的基本问题--多准则极大极小不等式的特邀报告。2002年6月，我们在日本奈良组织了第五届多目标规划和目标规划理论与应用国际会议（MOPGP'02），并在会上讨论了几个问题。2.在NACA 2001、MOPGP'02和NACA 2003上，我们分别给出了关于标量化函数从集值映射的凸性和不连续性继承的几个性质的结果。在NACA 2001、MOPGP'02和NACA 2003上给出了多目标博弈的结果。在NACA 2001和NACA 2003上给出了多目标选址问题的结果。这些结果发表在RIMS非线性分析和凸分析研讨会上。我们还在福井工业大学举行的日本运筹学学会“决策与OR”研究小组上发表了它们。

In this research, we have studied several kinds of themes on vector optimization analytically and broadly and then we got some results on saddle point's existence, vector equilibrium problem, convexity and semicontinuity for set-valued maps under systematic research and investigation. Moreover, we have studied on efficient solutions of multicriteria location problems. The main results and presentations are as follows.1.We presented our results on characterization of efficient solutions and existence of cone saddle points at The Second and Third International Conferences on Nonlinear Analysis and Convex Analysis(NACA2001,NACA2003), held at Hirosaki University and Tokyo Institute of Technology, August 2001 and August 2003, respectively. We delivered invited talks on multicriteria minimax inequality which is fundamental for cone saddle point of vector-valued function at NACA2003 and Workshop on Nonlinear Analysis & Convex Analysis, November 2003. Moreover, we organized the Fifth International Conference on Multi-Objective Programming and Goal Programming : Theory & Applications(MOPGP'02) at Nara, June 2002, and we discussed several problems and topics in this meeting.2.We presented our results on several inherited properties of scalarizing functions from convexity and semicontinuity of set-valued maps at NACA2001, MOPGP'02, and NACA2003. These results were presented in RIMS Symposium on Nonlinear Analysis and Convex Analysis.3.We presented our results on multicriteria game at NACA2001,MOPGP'02, and NACA2003.4.We presented our results on multicriteria location problems at NACA2001 and NACA2003. These results were presented in RIMS Symposium on Nonlinear Analysis and Convex Analysis. We also presented them at the research group of "decision and OR" of the Operations Research Society of Japan, held at Fukui University of Technology.