Characterizations Of The Solution Sets Of Generalized Convex Fuzzy Mathematical Programming

Recently, as an important research direction of Optimization theory and applications,fuzzy optimization theory have been developed rapidly and become one of the important research fields in Optimization. It has been playing an important role in actual life, such as μfuzzy control, fuzzy decision-making, systems theory, Biological Sciences, etc. Fuzzy mathematical programming is abstracted away from much real problem, a lot of equal form of characterizations of the optimal solution sets is an indispensable work. We devote to research generalized convexity of fuzzy mapping and characterizations of solution sets for fuzzy mathematical programming problems with generalized convexity. This article contain five chapters as followsμIn chapter 1, we describe the research background and status of fuzzy mathematical programming. And we also summarize the main content.In chapter 2, we introduce some basic concepts and prepared knowledge, including fuzzy sets and fuzzy numbers, convex fuzzy mapping and generalized convex fuzzy mapping, Fuzzy directional derivative, etc.In Chapter 3, we mainly study some properties of preinvex fuzzy mapping. Firstly,we give the necessary and su?cient condition of di?erentiable preinvex fuzzy mapping by concept of the di?erentiable fuzzy mapping, further more, it is built to the equivalent conditions of fuzzy mapping and positive semidefinite matrix. Next, we propose the notion of fuzzy η-Gateaux di?erential, also get some properties of fuzzy Gateaux di?erential and the fuzzy subdi?erential monotonicity, studying the relationship between fuzzy η-Gateaux di?erential and fuzzy subdi?erential.In Chapter 4, we study characterizations of the optimal solution sets for some fuzzy mathematical programming problem under the suitable generalized convexity.Firstly, we build some equivalent conditions by fuzzy η-Gateaux di?erential. Secondly,characterizations of the optimal solution sets for preinvex fuzzy mathematical programming problems under the condition of fuzzy η-Gateaux di?erential.In Chapter 5, we summarize the research work of this article, and the possible problems and further research are pointed out.