Duality theory in fuzzy mathematical programming problems with fuzzy coefficients

In this paper, the notions of subgradient, subdifferential, and differential with respect to convex fuzzy mappings are investigated, which provides the basis for the fuzzy extremum problem theory. We consider the problems of minimizing or maximizing a convex fuzzy mapping over a convex set and develop the necessary and/or sufficient optimality conditions. Furthermore, the concept of saddle-points and minimax theorems under fuzzy environment is discussed. The results obtained are used to formulate 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. Finally, these results are applied to fuzzy linear programming and fuzzy quadratic programming.     

Fixed point theorems on fuzzy normed linear spaces

In this paper, definitions of strongly fuzzy convergent sequence, l-fuzzy weakly convergent sequence and l-fuzzy weakly compact set are given in a fuzzy normed linear space. The concepts of fuzzy normal structure, fuzzy non-expansive mapping, uniformly convex fuzzy normed linear space are introduced and fixed point theorems for fuzzy non-expansive mappings are proved.     

Set-based Bayesianism

Problems for strict and convex Bayesianism are discussed. A set-based Bayesianism generalizing convex Bayesianism and intervalism is proposed. This approach abandons not only the strict Bayesian requirement of a unique real-valued probability function in any decision-making context but also the requirement of convexity for a set-based representation of uncertainty. Levi's E-admissibility decision criterion is retained and is shown to be applicable in the nonconvex case     

直觉模糊映射的凸分析

介绍了直觉模糊数和直觉模糊映射的凸性、凹性、拟凸性、拟凹性\上半连续性、下半连续性和正齐次性的定义.通过引入一种新的偏序关系来研究凸直觉模糊映射性质,对凸直觉模糊映射和凹直觉模糊映射分别建立了两个刻画定理.讨论了直觉模糊映射的凸性与拟凸性的关系,证明了一个凸直觉模糊映射必是一个拟凸直觉模糊映射和一个凹直觉模糊映射必是一个拟凹直觉模糊映射.考虑了直觉模糊映射的凸性与半连续性的关系,获得了直觉模糊映射的凸性与上半连续性(下半连续性)等价的条件.在直觉模糊算子方面,给出了正齐次直觉模糊映射是凸直觉模糊映射的充要条件及直觉模糊算子是凸直觉模糊映射的判别定理.拓展了经典集合上的凸函数和凸模糊映射的相关理论,使之成为直觉模糊理论的有益补充.     

Fuzzy linear regression model based on fuzzy scalar product

The new concept and method of imposing imprecise (fuzzy) input and output data upon the conventional linear regression model is proposed in this paper. We introduce the fuzzy scalar (inner) product to formulate the fuzzy linear regression model. In order to invoke the conventional approach of linear regression analysis for real-valued data, we transact the α-level linear regression models of the fuzzy linear regression model. We construct the membership functions of fuzzy least squares estimators via the form of “Resolution Identity” which is a well-known formula in fuzzy sets theory. In order to obtain the membership value of any given least squares estimate taken from the fuzzy least squares estimator, we transform the original problem into the optimization problems. We also provide two computational procedures to solve the optimization problems.     

Extended triangular norms

The paper is devoted to classical t-norms extended to operations on fuzzy quantities in accordance with the generalized Zadeh extension principle. Such extended t-norms are used for calculating intersection of type-2 fuzzy sets. Analytical expressions for membership functions of some extended t-norms are derived assuming special classes of fuzzy quantities, i.e., fuzzy truth intervals or fuzzy truth numbers. The possibility of applying these results in the construction of type-2 adaptive network fuzzy inference systems is illustrated on several examples.     

Semistrictly preinvex fuzzy mappings

In this paper, we introduce the concepts of a strictly preinvex fuzzy mapping, a semistrictly preinvex fuzzy mapping, a strictly prequasi-invex fuzzy mapping and a semistrictly prequasi-invex fuzzy mapping. Some properties of these mappings and relations among them are studied. A few interesting theorems are proved, including that: (1) a strictly prequasi-invex fuzzy mapping has at most one global minimum point; (2) the solution set is invex if the fuzzy mapping is prequasi-invex and (3) prequasi-invex fuzzy mappings satisfy most of the basic properties of the preinvex fuzzy mappings.     

The theoretical foundations of statistical learning theory based on fuzzy number samples

Statistical learning theory based on real-valued random samples has been regarded as a better theory on statistical learning with small sample. The key theorem of learning theory and bounds on the rate of convergence of learning processes are important theoretical foundations of statistical learning theory. In this paper, the theoretical foundations of the statistical learning theory based on fuzzy number samples are discussed. The concepts of fuzzy expected risk functional, fuzzy empirical risk functional and fuzzy empirical risk minimization principle are redefined. The key theorem of learning theory based on fuzzy number samples is proved. Furthermore, the bounds on the rate of convergence of learning processes based on fuzzy number samples are discussed.     

基于遗传算法的模糊控制在智能系统中的应用

本文针以吊车系统为例给出了一种智能模糊控制器即分段切换因子模糊控制器，并运用遗传算法进行参数调整，确保吊车系统的精度，仿真结果表明，该控制器精度高、抗干扰能力强、实时性好。     

基于FCM和RVGA的FNN在非线性辨识中的应用

提出一种新的基于模糊C-均值聚类(FCM)和实值遗传算法(RVGA)的模糊神经网络(FNN)。在对模糊规则进行训练之前,利用模糊C-均值聚类从训练数据中提取出典型数据,以删除野值和协调数据内部冲突。然后利用一种新的实值遗传算法对此典型数据进行训练。此遗传算法的交叉和变异运算均直接对实值进行操作,而不是传统的位操作,因此,可以极大地减少训练时间并实现全局寻优。对非线性函数辨识的仿真实验证明了该方法的优越性。     

A note on smooth approximation capabilities of fuzzy systems

Modeling and prediction in some systems requires the simultaneous approximation of mappings and their derivatives to a certain finite order. In this paper, universal approximation capabilities of fuzzy systems are extended to this situation, by showing the denseness of some general classes of fuzzy models in appropriate function spaces where distance between functions is defined in terms of their derivatives. Requirements are generally very mild, and are often limited to imposing sufficient differentiability conditions on the classes of fuzzy systems to be used. The cases covered by this paper include additive fuzzy systems with Gaussian membership functions and general additive fuzzy models with realistic membership functions. Some potential application fields are considered, including examples from economics and time series prediction     

Convex fuzzy mapping and operations of convex fuzzy mappings

Convex and concave fuzzy mappings are considered in this paper. Based on the seminal work of Nanda and Kar on the definition of convex fuzzy mapping, the operations of convex fuzzy mapping are established. Some of the important concepts such as positively homogeneous, infimal convolution, right scalar multiplication, and convex hull are introduced and the corresponding theorems are demonstrated by using the parametric representation of fuzzy numbers.     

Stability of vector optimization problems with fuzzy weights in the objective functions and fuzzy matrix parameters in the constraints

The aim of this paper is to investigate the stability of multiobjective nonlinear programming problems with fuzzy weights in the objective functions and fuzzy matrix parameters in the constraints and represent, in addition, the related dual problems for which the set of feasible parameters and the solvability set are studied. These fuzzy weights and fuzzy matrix parameters are characterized by fuzzy numbers. The existing results concerning the basic notions parametric space in convex programs are redefined and analyzed qualitatively under the concept of α-Pareto optimality. An illustrative example is given to clarify the obtained results.     

Techniques for Gradient-Based Bilevel Optimization with Non-smooth Lower Level Problems

We propose techniques for approximating bilevel optimization problems with non-smooth lower level problems that can have a non-unique solution. To this end, we substitute the expression of a minimizer of the lower level minimization problem with an iterative algorithm that is guaranteed to converge to a minimizer of the problem. Using suitable non-linear proximal distance functions, the update mappings of such an iterative algorithm can be differentiable, notwithstanding the fact that the minimization problem is non-smooth.     

Multiple Fuzzy Relation and Its Application to Coupled Fuzzy Control

Using semi‐tensor product (STP) of matrix, this paper investigates the fuzzy relation of multiple fuzzy and uses this to design coupled fuzzy control is designed. First of all, under the assumption that the universe of discourse is finite, a fuzzy logical variable can be expressed as a vector, which unifies the expression of elements, subsets, and fuzzy subsets of a universe of discourse. Then, the matrix expression of set mappings is naturally extended to fuzzy sets. Second, based on STP, logic‐based matrix addition and product are proposed. These are particulary useful for the calculation of compounded fuzzy relations. Third, a dual fuzzy structure is introduced, which assures the finiteness of the universe of discourse, and is used for fuzzification and defuzzification. Finally, using the results obtained, a new technique is developed to design a coupled fuzzy controller for multi‐input multi‐output (MIMO) systems with coupled multiple fuzzy relations.     

New approaches to relaxed stabilization conditions and H-infinity control designs for T-S fuzzy systems

This paper focuses on the problem of fuzzy control for a class of continuous-time T-S fuzzy systems.New methods of stabilization design and H infinity control are derived based on a relaxed approach in...     

Some results on fuzzy (digital) convexity

The definition of fuzzy convexity is reviewed and some results on projections of convex and fuzzy convex sets are established. Digital fuzzy convexity is defined and relationships among alternative definitions are investigated.     

Fuzzy random programming with equilibrium chance constraints

To model fuzzy random decision systems, this paper first defines three kinds of equilibrium chances via fuzzy integrals in the sense of Sugeno. Then a new class of fuzzy random programming problems is presented based on equilibrium chances. Also, some convex theorems about fuzzy random linear programming problems are proved, the results provide us methods to convert primal fuzzy random programming problems to their equivalent stochastic convex programming ones so that both the primal problems and their equivalent problems have the same optimal solutions and the techniques developed for stochastic convex programming can apply. After that, a solution approach, which integrates simulations, neural network and genetic algorithm, is suggested to solve general fuzzy random programming problems. At the end of this paper, three numerical examples are provided. Since the equivalent stochastic programming problems of the three examples are very complex and nonconvex, the techniques of stochastic programming cannot apply. In this paper, we solve them by the proposed hybrid intelligent algorithm. The results show that the algorithm is feasible and effectiveness.     

Differentiability and convexity of fuzzy mappings

Goetschel and Voxman [1] have introduced the notion of a derivative for fuzzy mappings of one variable in a manner different from the usual one. In this paper, we define a differentiable fuzzy mapping of several variables in ways that parallel the definition, proposed by Goetschel and Voxman [1], for a fuzzy mapping of one variable, and then study some basic differentiability properties of fuzzy mappings from the standpoint of convex analysis.