Characterizations Of The Solution Set Of Generalized Convex Fuzzy Optimization Problem And Vector Variational Inequalities

Fuzzy optimization is a kind of model and method to deal with optimization problems with uncertainty.The characterization of the solution set is not only conducive to understanding the structure of the solution of optimization problems with multiple optimal solutions,but also has important theoretical significance for designing various algorithms.We find that fuzzy convexity plays an important role in characterizations of the solution set of fuzzy optimization problems.However,we note that some fuzzy optimization problems do not satisfy the condition of fuzzy convexity.Therefore,this thesis aims at introducing some new types of generalized convexity of fuzzy functions and comparing them with those of existing literatures.Then we discuss the properties of some generalized convex fuzzy functions.Based on the above research,we study the characterization of the solution set of fuzzy optimization problem.Finally,we discuss the relations between fuzzy vector variational inequalities and fuzzy vector optimization problem.The main contents of this thesis are as follows.In chapter one,firstly,we recall the research progress of fuzzy generalized convexity.Secondly,we recall the current research situation on the characterizations of the solution set of optimization problems.Thirdly,we recall the progress of the fuzzy variational inequalities.Finally,we give the main contents studied.In chapter two,we recall some basic definitions and related theories of fuzzy mathematics,including operation of fuzzy numbers,fuzzy order relations,fuzzy mapping,fuzzy differentiability,convex fuzzy functions,preinvex fuzzy functions and their equivalent characterizations.In chapter three,firstly,we define ?-preinvexity and ?-prequasiinvexity of fuzzy functions and give an equivalent characterization of the fuzzy ?-preinvex function.Some examples are given to describe these generalized convex fuzzy functions.Secondly,we use the H-difference of fuzzy numbers to give the ??-directional derivatives of a fuzzy function,and propose the ?-pseudoinvex convexity and ??-pseudomonotonicity of fuzzy functions with the help of the ??-directional derivatives.Some examples are given to illustrate these definitions.Finally,by applying the g-differentiability of fuzzy functions,we introduce some more general fuzzy generalized convexity,such as the ?-invexity of fuzzy functions.In chapter four,we study characterizations of the solution set of fuzzy optimization problem.Firstly,we propose a new condition and give an example to show its existence.Secondly,under the condition of fuzzy radial semi-continuity and other suitable conditions,we give a necessary and sufficient condition of fuzzy ?-preinvexity,prove the fuzzy ?-pseudoinvexity is equivalent to the fuzzy ??-pseudomonotonicity and discuss the properties of other generalized convex fuzzy functions.Finally,by using these properties of generalized convex fuzzy functions,we study characterizations of the solution set of non-differentiable ?-pseudoinvex fuzzy optimization problem.In chapter five,we introduce the fuzzy vector variational inequalities.Under the assumptions of ?-invexity,?-strict incavity and ??-pseudoinvexity of g-differentiable fuzzy vector functions,we discuss the relations between the solution of fuzzy variationallike inequality and the solution of fuzzy optimization problem.