Fuzzy Convex Research Articles

Geometric properties of ternary fuzzy relations

Ternary fuzzy relations, and fuzzy betweenness relations in particular, are witnessing increasing attention in recent years. A key reason is that axiomatic properties of ternary fuzzy relations seem to be ideally suited to capture geometric characteristics of the abstract notion of betweenness. In this paper, we introduce several new properties of ternary fuzzy relations, including the Peano property, the Pasch property and the sand-glass property, that can be qualified as geometric properties. We investigate their interrelationships as well as their connections with various types of fuzzy betweenness relations. Additionally, in the context of our study of the Pasch property and the sand-glass property, we introduce the convexity property of ternary fuzzy relations by taking inspiration from the solid theoretical basis of the theory of fuzzy convex structures.

A New Neurodynamics-Based Model for Fuzzy Convex Optimization Problems With Fuzzy Coefficients and General Constraints

A New Neurodynamics-Based Model for Fuzzy Convex Optimization Problems With Fuzzy Coefficients and General Constraints

A Recurrent Neural Network Approach for Constrained Distributed Fuzzy Convex Optimization.

This article investigates a class of constrained distributed fuzzy convex optimization problems, where the objective function is the sum of a set of local fuzzy convex objective functions, and the constraints include partial order relation and closed convex set constraints. In undirected connected node communication network, each node only knows its own objective function and constraints, and the local objective function and partial order relation functions may be nonsmooth. To solve this problem, a recurrent neural network approach based on differential inclusion framework is proposed. The network model is constructed with the help of the idea of penalty function, and the estimation of penalty parameters in advance is eliminated. Through theoretical analysis, it is proven that the state solution of the network enters the feasible region in finite time and does not escape again, and finally reaches consensus at an optimal solution of the distributed fuzzy optimization problem. Furthermore, the stability and global convergence of the network do not depend on the selection of the initial state. A numerical example and an intelligent ship output power optimization problem are given to illustrate the feasibility and effectiveness of the proposed approach.

Binary convex fuzzy vector spaces over binary vector spaces and their applications

Binary convex fuzzy vector spaces over binary vector spaces and their applications

Connectedness degrees in (L, M)-fuzzy convex structures

Connectivity is one of the most essential notions in general topology. Convex structures are topological-like structures. Many properties in topological spaces have been generalized to convex structures, such as separation. However, connectivity has not been studied in convex structures yet. In this paper, firstly, based on the consideration to hull operators, separatedness is defined in classical convex structures, and then we provide the concept of connectivity. Secondly, some equivalent characterizations of connectivity are discussed, and we investigate the related properties of connectivity. In additional, through (L, M)-fuzzy convex hull operators, we propose the separatedness degrees of (L, M)-fuzzy convex structures. Furthermore, the notion of connectedness degrees of (L, M)-fuzzy convex structures is introduced. Finally, many properties of connectivity in general convex structures can be generalized to (L, M)-fuzzy convex structures.

Solving Fuzzy Convex Programming Problems via a Projection Neural Network Framework

In this paper, the solution of the fuzzy nonlinear optimization problems (FNLOPs) is given using a projection recurrent neural network (RNN) scheme. Since there is a few research for resolving of FNLOP by RNNs, we describe a new framework to solve the problem. By reformulating the original program to an interval problem and then weighting problem, the Karush–Kuhn–Tucker (KKT) conditions are obtained. Moreover, we utilize the KKT conditions into a RNN as a capable tool to solve the problem. Besides, the global convergence and the Lyapunov stability of the neuro-dynamic model are established. In the final step, some simulation examples are stated to validate the obtained results. Reported results are compared with some other previous neural networks.

Arithmetic Mean-Geometric Mean Inequality for Convex Fuzzy Sets

Convex analysis is a discipline of mathematics dedicated to the explication of the properties of convex sets and convex functions. Convex functions are extremely useful in proving many famous inequalities in mathematics. It is widely known that inequalities defined by convex functions originated with the works done by Holder and Jensen among others and applied to modeling a variety of problems both in hard and soft sciences. It is said that the arsenal of an analyst is heavily stocked with inequalities. Studies related to convexity have kept occupying a central position in almost all areas of mathematics, especially in functional analysis and operations research. Following the emergence of fuzzy set theory, fuzzy convexity alongside a number of related concepts has been explicated. However, not much has been done regarding fuzzy inequalities defined by fuzzy convexity. In this paper, a novel attempt to study arithmetic mean-geometric mean is proposed. In this short note, we provide two proofs of the arithmetic mean-geometric mean inequality for convex fuzzy sets. Mathematics Subject Classification 2020: 03E72, and 94D05.

A new characterization of the product of (L, M)-fuzzy convex structures1

In order to give a characterization of the product of (L, M)-fuzzy convex structures, the notion of convex (L, M)-fuzzy hull operators is presented, it is proved that the category of (L, M)-fuzzy convex structures and the category of convex (L, M)-fuzzy hull operators are isomorphic. In particular, the lattices structure of convex (L, M)-fuzzy hull operators and a new characterization of the product of (L, M)-fuzzy convex structures are given.

A Fuzzy Convex Nonparametric Least Squares Method with Different Shape Constraints

A Fuzzy Convex Nonparametric Least Squares Method with Different Shape Constraints

Hermite–Hadamard–Type Fractional–Integral Inequalities for (p,h)–Convex Fuzzy–Interval–Valued Mappings

Hermite–Hadamard–Type Fractional–Integral Inequalities for (p,h)–Convex Fuzzy–Interval–Valued Mappings

Product and coproduct structures of (L, M)-fuzzy hull operators1

Notice that there’s a one-to-one correspondence between between (L, M)-fuzzy hull operators and (L, M)-fuzzy convex structures. So, it is necessary to consider the product of (L, M)-fuzzy convex structures through the product structures of (L, M)-fuzzy hull operators. In this paper, we construct the product structures of (L, M)-fuzzy hull operators to characterize the product of (L, M)-fuzzy convex structures. On the other hand, we construct the coproduct structures of (L, M)-fuzzy hull operators to give a reasonable definition with respect to the coproduct of (L, M)-fuzzy convex structures.

The KKT Optimality Conditions and Duality for Constrained Programming Problem with Generalized α- Convex Fuzzy Functions

This paper mainly studies the mixed constraint interval programming problem under the generalized convex fuzzy mapping. Firstly, this paper give the concepts of fuzzy mappings, such as quasiconvex, strictly quasiconvex, pseudoconvex and strictly pseudoconvex. Then, the relation of generalized convex fuzzy mapping is studied and some properties are obtained. Finally, the necessary and sufficient KKT conditions are given, and the duality problem is established. The weak duality, strong duality and inverse duality theorem of fuzzy interval programming are proved.

Convergence structures in (L,M)-fuzzy convex spaces

This paper presents the concepts of (L,M)-remotehood spaces and (L,M)-convergence spaces in the framework of (L,M)-fuzzy convex spaces. Firstly, it is shown that the category of (L,M)-remotehood spaces is isomorphic to the category of (L,M)-fuzzy convex spaces. Secondly, it is proved that the category of (L,M)-fuzzy convex spaces can be embedded in the category of (L,M)-convergence spaces as a reflective subcategory. Finally, the concepts of preconvex (L,M)-remotehood spaces and preconvex (L,M)- convergence spaces are introduced and it is shown that the category of preconvex (L,M)-remotehood spaces is isomorphic to the category of preconvex (L,M)-convergence spaces.

Fixed Point Theorems in $\mathscr{G}$ - Fuzzy Convex Metric Spaces

This work introduces a new three-step iteration process and shows that the same leads to a unique fixed point with the help of theorems under different conditions of contractive mappings over-generalized $\mathscr{G}$ - fuzzy metric spaces in the convex structure. Also, we investigate the data dependence result of this iterative process in the generalized $\mathscr{G}$ - fuzzy convex metric spaces.

STUDY OF CONVEXITY IN SETS AND FUZZY SETS

In this paper, we present an overview of both classical and fuzzy convexity, particularly, in conjunction with continuity5 and some topological concepts, and provide proofs of some of their algebraic properties along with suitable illustrations. We have extended the work of (Ammar E. E., 1999) by proving that if and are two convex fuzzy sets, then , , , and are also convex fuzzy sets. We have also shown that the convexity of a fuzzy set implies semistrict quasi-convexity

The topological structure of the set of fuzzy numbers with Lp metric

For a non-degenerate convex subset Y of the n-dimensional Euclidean space Rn, let K(Y) be the family of all fuzzy sets in Rn, which are upper-semicontinuous, fuzzy convex, normal and with compact supports included in Y. We show that the space K(Y) with the topology of the Lp metric with p≥1, that is, for f,g∈K(Y),Lp(f,g)=(∫01(dH([f]α,[g]α))pdα)1p, is homeomorphic to the Hilbert cube Q=[−1,1]ω if and only if Y is compact; and the space K(Y) is homeomorphic to the pseudoboundary B(Q)=Q∖(−1,1)ω of the Hilbert cube Q=[−1,1]ω if and only if Y is non-compact and locally compact.

Complex intuitionistic fuzzy soft lattice ordered group associated with ℓ-ideal

In this paper, we study the concept of complex intuitionistic fuzzy soft convex ℓ-group . We examine some of its innate algebraic properties theoretically. Moreover, we define the notion of complex intuitionistic fuzzy soft ℓ-ideal . We analyze the relationship between the , structures and corresponding structures in complex intuitionistic fuzzy soft lattice ordered group .

L-topological derived neighborhood relations and L-topological derived remotehood relations

Fuzzy relations and fuzzy derived operators are useful tools to characterize fuzzy mathematical structures such as fuzzy topology, fuzzy convexity, fuzzy matroid and fuzzy convergence structure. In this paper, notions of L-topological neighborhood relation space, L-topological derived neighborhood relation space and L-topological derived neighborhood space are introduced. It is proved that all of these spaces are categorically isomorphic to L-topological internal relation space and L-topological neighborhood space. Also, notions of L-topological remotehood relation space, L-topological derived remotehood relation space and L-topological derived remotehood space are introduced. It is proved that all of these spaces are categorically isomorphic to L-topological enclosed relation space and L-topological remotehood space.

(L, M)- fuzzy (restricted) convex derived hull spaces

In this paper, the notion of (L, M)- fuzzy convex derived hull spaces is introduced. It is proved that the category of (L, M)- fuzzy convex derived hull spaces is isomorphic to the category of (L, M)- fuzzy convex spaces and the category of (L, M)- fuzzy convex enclosed relation spaces. Based on this, the notion of (L, M)- fuzzy restricted convex derived hull spaces is introduced. It is further proved that the category of (L, M)- fuzzy restricted convex derived hull spaces is isomorphic to the category of (L, M)- fuzzy restricted convex hull spaces.

Fuzzy Detour Convexity and Fuzzy Detour Covering in Fuzzy Graphs

A path P connecting a pair of vertices in a connected fuzzy graph is called a fuzzy detour, if its μ - length is maximum among all the feasible paths between them. In this paper we establish the notion of fuzzy detour convex sets, fuzzy detour covering, fuzzy detour basis, fuzzy detour number, fuzzy detour blocks and investigate some of their properties. It has been proved that, for a complete fuzzy graph G, the set of any pair of vertices in G is a fuzzy detour covering. A necessary and sufficient condition for a complete fuzzy graph to become a fuzzy detour block is also established. It has been proved that for a fuzzy tree there exists a nested chain of sets, where each set is a fuzzy detour convex. Application of fuzzy detour covering and fuzzy detour basis is also presented.