Optimality conditions for invex nonsmooth optimization problems with fuzzy objective functions

In this paper, the definitions of Clarke generalized directional alpha -derivative and Clarke generalized gradient are introduced for a locally Lipschitz fuzzy function. Further, a nonconvex nonsmooth optimization problem with fuzzy objective function and both inequality and equality constraints is considered. TheKarush-Kuhn-Tucker optimality conditions are established for such a nonsmooth extremum problem. For proving these conditions, the approach is used in which, for the considered nonsmooth fuzzy optimization problem, its associated bi-objective optimization problem is constructed. The bi-objective optimization problem is solved by its associated scalarized problem constructed in the weighting method. Then, under invexity hypotheses, (weakly) nondominated solutions in the considered nonsmooth fuzzy minimization problem are characterized through Pareto solutions in its associated bi-objective optimization problem and Karush-Kuhn-Tucker points of the weighting problem.