高维模糊数值函数分析学、模糊凸分析与优化理论

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.

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。

本项目对高维模糊数值函数分析学、模糊凸分析与优化理论进行了系统研究。主要包括高维模糊数值函数积分理论，高维模糊数值函数可微性问题和Radon-Nikodym性质，高维模糊数值函数的积分表示以及积分与微分的相互关系，高维模糊数值函数的凸性，高维模糊数值函数凸优化的KKT条件和模糊数值函数的变分不等式等。在研究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，利用经典实分析、凸分析、集值分析、泛函分析理论，研究了基于模糊 Henstock 积分的模糊Laplace 变换，Vitali覆盖意义下模糊数值函数Henstock积分的控制收敛定理，高维模糊数空间上的偏序关系，高维模糊映射的凸分析和模糊优化理论，高维模糊数值函数的广义微分及其凸优化问题，模糊 Lagrange函数的鞍点问题和鞍点规划，高维方模糊数值函数的广义微分及其模糊约束优化问题的KKT优化条件，模糊数向量值函数的似变分不等式问题等。同时，研究了基于非可加测度的模糊分析学，基于模糊数值函数的不适定问题及其正则化方法，模糊数列和模糊数值函数列空间结构理论，以及几类涉及模糊数的完全离散信息系统。