Fuzzy Convex Analysis And Its Applications To Fuzzy Programming

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.