Optimality conditions for fuzzy constrained programming problems

Abstract

This paper solves optimization problems where both the objective and constraints are given by fuzzy functions. In order to get it, we first prove that these problems are equivalent to optimization problems where the constraints functions are non-fuzzy functions and we introduce a new and wider stationary point concept that generalizes all existing concepts so far. This new stationary point concept is based on the gH-differentiability and has many computational advantages that we describe. It is well-known that obtain a useful differentiability notion for fuzzy functions is a difficult task without linearity. And we are in that case due to the fact that the fuzzy numbers (intervals) space is a nonlinear one. In this direction, the gH-derivative for fuzzy functions is a concept that is more general than Hukuhara and level-wise derivatives that are usually used in fuzzy optimization so far, in the sense that they can be applied to a wider number of fuzzy function classes than above concepts. With this new differentiability concept, we prove a necessary optimality condition for fuzzy optimization problems that is more operational and less restrictive that the few ones we can find in the literature so far. Moreover, due to the fact that we do not have a linear space for fuzzy numbers, the convex concepts and generalized convex fuzzy function notion are very restrictive, also. This implies that the sufficiency optimality conditions for fuzzy problems published so far are not useful.