模糊映射的次可微性及其应用

项目名称： 模糊映射的次可微性及其应用

项目编号： No.11461052

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： 包玉娥

作者单位： 内蒙古民族大学

项目金额： 37万元

中文摘要： 随着模糊规划研究的深入与发展，许多学者开始研究了模糊映射的次微分理论及应用问题，并得到了一些重要结论。但由于模糊数的基于区间数的偏序关系条件较强，导致非光滑分析中的很多结论在模糊数空间中无法表示和讨论。所以有必要选择或建立合理的模糊数的序关系，讨论模糊映射的次微分及应用问题。基于这种想法，本申请项目拟用Goetschel-Voxmann 所定义的序关系，（1）讨论一般模糊映射的次可微性问题，给出模糊映射的变分原理，建立近似和规则；（2）给出模糊映射的共轭映射概念，建立次微分与共轭映射之间的关系式，并用共轭映射讨论凸模糊极值问题的对偶性和稳定性，给出相关的基本结论；（3）讨论模糊意义下的鞍点与极小极大定理，并结合模糊规划的Lagrange对偶，建立凸模糊规划的Lagrange对偶和KKT条件；⑷ 将获得的有关结论应用于模糊二次规划的研究中，并研究在生产系统、工程技术中具有模糊信息的应用案例。

中文关键词： 模糊数学；模糊集；模糊拓扑；模糊推理；模糊逻辑

英文摘要： With in-depth study and development of the fuzzy programming, many scholars begin to study the subdifferential theory of fuzzy mapping and its applications and obtain some important results. But the conditions of partial order based on the interval number for fuzzy numbers is so strong that many conclusions of non-smoothing analysis cannot be represented and discussed in fuzzy numbers space. So the reasonable order relations of fuzzy numbers should be selected or established, then the subdifferential of fuzzy mapping and its applications should be discussed. In this project, using the order relation defined by Goetschel-Voxmannb, we will do following work: (1) we will discuss the subdifferentiability of general fuzzy mapping , and give the variational principle of fuzzy mapping, and establish the approximation sum rules; (2) we will give the concept of conjugate mapping of fuzzy mapping, and establish the relationship between subdifferential and conjugate mapping, and discuss the duality and stability of convex fuzzy extremum by using conjugate mapping, and obtain some related results; (3) we will discuss the saddle point and min-max theorems in the sense of fuzzy, and establish Lagrange duality and KKT conditions of convex fuzzy programming by Lagrange duality of fuzzy programming; (4) we will apply the related results to the research of quadratic programming, and study the application cases in production systems and engineering technology with fuzzy information.

英文关键词： fuzzy mathmematics;fuzzy set;fuzzy topology;fuzzy inference;fuzzy logic