【作者】 张成；

【导师】 夏尊铨；

【作者基本信息】 大连理工大学 ， 运筹学与控制论， 2004， 博士

【摘要】 本文系统地研究了模糊凸分析与模糊优化及其它们之间的联系。在研究模糊向量空间、模糊凸集和模糊凸映射的基础上，我们研究了模糊规划的Lagrange对偶和凸模糊规划的KKT条件，并将有关结果应用到模糊线性规划和模糊二次规划的研究中。 取得的主要结果可概括如下： 1 在第3章中，从随机落影理论和“模糊直线”的角度研究模糊向量空间；给出了模糊仿射变换的概念，研究了模糊仿射变换与模糊线性变换之间的关系；引入了反模糊数的概念，并给出了有关的基本性质；利用三角范数研究了模糊内积空间；最后，讨论了模糊向量的内积。 2 在第4章中，建立了有关凸模糊映射的理论：建立了关于凸模糊映射的Jensen不等式、模糊正齐次映射、凸模糊映射的下卷积、右数乘和凸包等概念，利用模糊数的参数化表示，给出了相应的定理；在反模糊数空间，对凸模糊映射的共轭也作了探讨，证明了凸模糊映射的共轭集合和共轭映射都是凸的；最后对凸模糊映射的次梯度、次微分和微分等概念进行了研究，为模糊极值理论打下了基础。 3 在第5章中，首先，在第4章所建立模糊凸分析的结构基础上，考虑了凸模糊映射的极值问题，得到了凸模糊映射取得极值的充分/必要条件。其次，讨论了模糊映射的鞍点与极小极大定理，并与模糊规划的Lagrange对偶联系起来，由此得到凸模糊规划的Lagrange对偶和KKT条件，并对”受扰”的凸模糊规划也作了讨论。最后，将所得到的结果应用到模糊线性规划与模糊二次规划的研究中。更多还原

【Abstract】 This dissertation studies systematically fuzzy convex analysis and fuzzy optimization and the relationships between them. Based on the theory of fuzzy vector subspaces, fuzzy convex sets and fuzzy convex mappings, the Lagrangian duality theory is studied and KKT conditions for fuzzy programming are derived. The results obtained for fuzzy convex programming are employed to the study of fuzzy linear programming and fuzzy quadratic programming.The main results obtained in this dissertation are summarized as follows:1. In chapter 3, the fuzzy vector subspaces are discussed from the views of random shadows and fuzzy lines; the notion of fuzzy affine transformation is introduced, and the relations between a fuzzy affine transformation and a fuzzy linear mapping are discussed. The concept of anti-fuzzy number is proposed, and several basic properties are presented. Fuzzy inner product spaces are investigated by T-norm and T-conorm, and the inner product of fuzzy vectors is concerned.2. Chapter 4 establishes the theory of convex fuzzy mappings: The concepts, such as Jensen’s inequality, positively homogeneous, infimal convolution, right scalar multiplication and convex hull are introduced. The corresponding theorems are demonstrated by using the parametric representations of fuzzy numbers. In anti-fuzzy number space, the conjugate mapping of convex fuzzy mapping is concerned, and convexities of conjugate set and conjugate mapping of convex fuzzy mapping are proved. The notions of subgradient, subdifferential, differential with respect to convex fuzzy mappings are investigated, which provides the basis of the theory of fuzzy extremum problems.3. Chapter 5 is devoted the study of fuzzy optimization in fuzzy convexanalysis structure given in chapter 4. We consider the problems of minimizing and maximizing a convex fuzzy mapping over a convex set and develop necessary and/or sufficient optimality conditions. We discuss the concept of saddle-points and minimax theorems under fuzzy environment. The results obtained are used to the Lagrangian dual of fuzzy programming. Under certain fuzzy convexity assumptions, KKT conditions for fuzzy programming are derived, and the "perturbed" convex fuzzy programming is considered. Furthermore, the above results are applied to fuzzy linear programming and fuzzy quadratic programming.更多还原