【摘要】 向量优化是数学规划一个重要分支,其理论与方法不仅与很多学科有密切联系,而且在新兴的多学科交叉领域中有着广泛的应用.本文从向量值广义凸映射、择一定理、线性标量化方法和Lagrange乘子存在性定理等4个方面对这一领域的研究进展情况及所用方法作了较为系统的总结.首先,介绍基于像空间方法的一类广义凸向量值映射和集值映射,总结已有的广义凸映射之间的关系.其次,介绍线性系统下择一定理到非线性系统下择一定理的发展,重点总结凸性或广义凸性条件下的择一定理研究.同时,针对择一定理的应用,给出向量优化问题各种解在凸或广义凸性条件下的线性标量化方法,进而总结向量优化问题的解,特别是真有效解的Lagrange乘子存在性结果.更多还原

【Abstract】 Vector optimization is an important part of mathematical programming. Its theory and methods have a promising interdisciplinary research field with many significant applications. In this survey, we mainly introduce the progress on the generalized convexity of vector-valued maps, alternative theorems, linear scalarization methods and Lagrange multiplier rules. We first introduce a class of generalized convexity for the vector-valued and setvalued maps, which is based on image space analysis, and summarize the relationship among them. Secondly,we introduce the development of the alternative theorems in linear systems to nonlinear systems. For nonlinear systems, we focus on the research of the alternative theorem under convexity or generalized convexity assumptions.Its applications in the linear scalarization and the Lagrange multiplier rules on vector optimization problems are summarized.更多还原