Abstract
Abstract The existing results on the variational inequality problems for fuzzy mappings and their applications were based on Zadeh’s decomposition theorem and were formally characterized by the precise sets which are the fuzzy mappings’ cut sets directly. That is, the existence of the fuzzy variational inequality problems in essence has not been solved. In this paper, the fuzzy variational-like inequality problems is incorporated into the framework of n-dimensional fuzzy number space by means of the new ordering of two n-dimensional fuzzy-number-valued functions we proposed [Fuzzy Sets and Systems 295 (2016) 19-36]. As a theoretical basis, the existence and the basic properties of the fuzzy variational inequality problems are discussed. Furthermore, the relationship between the variational-like inequality problems and the fuzzy optimization problems is discussed. Finally, we investigate the optimality conditions for the fuzzy multiobjective optimization problems.
Highlights
Variational inequality theory, where the function is a vector-valued mapping, known either in the form presented by Hartman and Stampacchia [1] or in the form introduced by Minty [2], has become an e ective and powerful tool for studying a wide class of linear/nonlinear problems arising in diverse applied elds such as optimization and control, mechanics, economics and engineering sciences
The fuzzy variational-like inequality problems is incorporated into the framework of n-dimensional fuzzy number space by means of the new ordering of two n-dimensional fuzzy-number-valued functions we proposed [Fuzzy Sets and Systems 295 (2016) 19-36]
In order to deal with the variational inequalities derived from some fuzzy environments, in 1989, Chang and Zhu [10] introduced the concepts of variational inequalities for fuzzy mapping in abstract spaces and investigated the existence of some types
Summary
Variational inequality theory, where the function is a vector-valued mapping, known either in the form presented by Hartman and Stampacchia [1] or in the form introduced by Minty [2], has become an e ective and powerful tool for studying a wide class of linear/nonlinear problems arising in diverse applied elds such as optimization and control, mechanics, economics and engineering sciences. Vector variational inequality, where the function is a matrix-valued mapping, was rst introduced and studied by Giannessi [3] in nitedimensional Euclidean spaces. This is a generalization of a scalar variational inequality to the vector case by virtue of multi-criteria considering. In order to deal with the variational inequalities derived from some fuzzy environments, in 1989, Chang and Zhu [10] introduced the concepts of variational inequalities for fuzzy mapping in abstract spaces and investigated the existence of some types
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