高维模糊数值函数分析学、模糊凸分析与优化理论

项目名称： 高维模糊数值函数分析学、模糊凸分析与优化理论

项目编号： No.11461062

项目类型： 地区科学基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 巩增泰

作者单位： 西北师范大学

项目金额： 36万元

中文摘要： 高维模糊数值函数分析学、模糊凸分析与优化理论包括：高维模糊数值函数积分理论及其数值计算、高维模糊数值函数可微性问题和Radon-Nikodym性质、高维模糊数值函数凸分析基础理论与若干模糊优化问题等。本项目在深入研究经典实分析、凸分析、集值分析、泛函分析及其与模糊数值函数分析学、模糊凸分析已有结果和相互联系的基础上，利用模糊数紧凸集表示定理、嵌入定理以及拟研究的与其他不确定集的转化定理，系统研究高维模糊数值函数积分理论及其数值计算、高维模糊数值函数可微性问题和Radon -Nikodym性质，得到高维模糊数值函数的积分表示以及积分与微分的相互关系；在定义和讨论高维模糊数值函数的凸性、建立高维模糊数值函数的不动点理论及FKKM定理和KyFan极大极小不等式、研究高维模糊数值函数的变分不等式的基础上，研究高维模糊数值函数在鞍点问题、数学规划、Nash平衡问题以及竞争均衡模型。

中文关键词： 模糊数；模糊数值函数；模糊分析学；模糊凸分析

英文摘要： The main reseach contents of the n-dimension fuzzy-number-valued functions analysis, fuzzy covex analysis and its optimization theory include the integral theory of the n-dimension fuzzy-number-valued functions and its numerical calculation, the differentiability of the n-dimension fuzzy-number-valued functions and its Radon-Nikodym theorems, the convex analysis of the n-dimension fuzzy-number-valued functions and its applications in fuzzy optimization problems. By analysing the previous results and the relationships of the real analysis, convex analysis, set-valued analysis, functional analysis, fuzzy analysis, fuzzy convex analysis, the compact convex representation theorem of the fuzzy numbers, the embedding theorems of the fuzzy numbers and the transformation theorem of the fuzzy numbers with some uncertainty sets, the integral theory of the n-dimension fuzzy-number-valued functions and its numerical calculation, the differentiability of the n-dimension fuzzy-number-valued functions and its Radon-Nikodym theorems are investigated systemly. As well as the integral representation, the relationship between the integral primitive and its integrand function are discussed. In addition, the convex analysis of the n-dimension fuzzy-number-valued functions are studied, such as the convexity, the fixed point theory, FKKM theorem and KyFan mini-max inequality, variational inequalities, and so on. Finally, some optimization theory and models are developed under the uncertainty circumstances which deal with the saddle point problems, mathematical programmings, Nash equilibrium problems and equilibrium competitive models.

英文关键词： Fuzzy Numbers;Fuzzy-number-valued Functions;Fuzzy Analysis;Fuzzy Convex Analysis