The Convexity And Differentiability Of N-Dimensional Fuzzy Mappings And Fuzzy Convex Optimization Theory

The research of convex analysis is closely related to the development of optimization theory.In mathematics,the critical importance is whether a mathematical programming problem can be transformed into a convex optimization model.However,a number of real world optimization problems often involve uncertain or imprecise data due to measurement errors or some uncertain factors.Fuzzy optimization was developed for dealing with real world problems where the problems are usually vague and not well defined.The fuzzy models not only avoid useful information and data loss,but also improve the flexibility and the operability of the model.There are numerous studies on the fuzzy convex analysis and its corresponding fuzzy optimization problems,but all the related works are only one-dimensional fuzzy-number-valued functions.Up to now,the systematic study on fuzzy convex optimization of n-dimensional fuzzy mappings did not see the report.The main reason is that there is almost no related research about the ordering and the difference of n-dimensional fuzzy numbers.The problems of the convexity and differentiability of n-dimensional fuzzy mappings and the corresponding convex optimization are investigated and discussed in detail based on the ordering and the difference of n-dimensional fuzzy numbers established in the paper.Firstly,the generalized difference of n-dimensional fuzzy numbers is characterized by using the support function and the generalized difference of n-dimensional fuzzy numbers is obtained.Taking into account that n-cell fuzzy numbers can provide more flexibility and tractability to represent the imprecise information,the generalized difference of n-cell fuzzy numbers is investigated and the representation of the level cut set for the generalized difference are given by the characterization theorem.We devote to seek an approximation operator that produces a 2-cell fuzzy number which is the nearest one to the given 2-dimensional fuzzy number,and the approximation operator preserves the centroid of the core with respect to the weighted distance.Secondly,the convexity of n-dimensional fuzzy mappings is discussed in detail,which is based on the partial ordering of n-dimensional fuzzy numbers proposed in the paper.The convexity,generalized convexity,lower and upper semicontinuity of n-dimensional fuzzy mappings are proposed some relations among the convexity and generalized convex-ity of n-dimensional fuzzy mappings and some criteria for the convexity and generalized convexity are obtained under the condition of the lower and upper semicontinuity are investigated by means of the convexity of vector valued mappings and the properties of n-dimensional fuzzy mappings.The results can be applied to fuzzy convex optimization theory directly and it is also pointed out that a local minimum point of a convex fuzzy mapping is a global minimum point of it.Thirdly,the differentiability for n-dimensional fuzzy-number-valued functions and n-dimensional fuzzy mappings are investigated in detail by utilizing the generalized dif-ference of n-dimensional fuzzy numbers.The Newton-Leibniz formula for a special type of n-cell fuzzy-number-valued functions (?)(t)= f(t)· u is given by means of the Riemann integral for n-dimensional fuzzy-number-valued functions.We put forward the concepts of differential and gradient of fuzzy mappings from Rm into En as well as discuss the characterizations of gradient by using the gradient of a crisp functions (?)(t)*(r,x)that are determined by the fuzzy mapping.Furthermore,based on the differentiability of the mapping f(t):M? R,the differentiability of a special type of n-cell fuzzy mapping (?)(t)= f(t)·u is characterized.Finally,by means of the concepts of differentiability and convexity of a fuzzy map-ping the constrained fuzzy convex optimization problem is discussed as well as the KKT optimality conditions for the constrained fuzzy convex optimization problem is obtained.Specifically,the fuzzy convex optimization with n-cell fuzzy mappings can be transformed into the classical convex optimization problem and give an example.