On optimality for fuzzy optimization problems with granular differentiable fuzzy objective functions - Antczak, Tadeusz (Expert systems with applications, 2024)
In recent years, there has been a growing interest in optimization problems with uncertainty. Fuzzy optimization is one of the approaches for investigation of such real-world extremum problems that contain uncertain data and, therefore, that are not well defined. Nonetheless, there is not enough discussion on the optimality conditions for optimal solutions (with regard to distinct fuzzy numbers) in (nonconvex) granular differentiable fuzzy optimization problems and methods for solving such uncertain extremum problems. Although the convexity notion is a very important property of optimization models, there are many real-world extremum problems and processes with uncertainty that cannot be modeled as convex fuzzy optimization problems. In this work, therefore, a new notion of granular differentiable generalized convexity is defined. Namely, the definition of granular differentiable F-convexity is introduced for fuzzy optimization problems and some properties of this concept are analyzed. Further, both the Fritz John necessary optimality conditions and, under the Guignard constraint qualification, the Karush-Kuhn-Tucker necessary optimality conditions are established for an optimal solution (with respect to distinct fuzzy numbers) in a fuzzy extremum problem with granular differentiable fuzzy-valued objective function and both inequality and equality constraints. As main applications of the introduced concept of granular differentiable F-convexity, the sufficiency of the derived Karush-Kuhn-Tucker necessary optimality conditions are proved and a new approach based on the so-called modified F-objective function method is proposed for solving the investigated fuzzy extremum problem.