Optimality results for a class of nonconvex fuzzy optimization problems with granular differentiable objective functions

There is the growing use in practice of optimization models with uncertain data related to human activity in which hypotheses are not verified in a way specific for classical optimization. Fuzzy optimization problems have been introduced and developed for formulating and solving such real-world extremum problems which are usually not well defined. In most works devoted to fuzzy optimization problems, fuzzy numbers are characterized by their vertical membership functions which causes some difficulties in calculations and is the reason for arithmetic paradoxes. In the paper, therefore, fuzzy numbers are characterized by their horizontal membership functions and the concept of a gr-derivative of a fuzzy function is used which is based on the horizontal membership function and the granular difference. Although the convexity notion is a very important property of optimization models, there are real-world processes and systems with uncertainty that cannot be modeled with convex fuzzy optimization problems. Therefore, new concepts of granular generalized convexity notions, that is, the concepts of granular pre-invexity and gr-differentiable invexity are introduced to fuzzy analysis and some properties of the aforesaid granular generalized convexity concepts are investigated. Further, the class of nonconvex smooth optimization problems with gr-differentiable fuzzy-valued objective function and differentiable inequality constraint functions is considered as an application of the concept of gr-differentiable invexity. Then, the Karush-Kuhn-Tucker necessary optimality conditions are established for a global fuzzy minimizer with regard to the distinct fuzzy numbers in the analyzed fuzzy extremum problem. Further, the sufficiency of the aforesaid necessary optimality conditions of a Karush-Kuhn-Tucker type is also proved.