Karush–Kuhn–Tucker conditions for interval and fuzzy optimization in several variables under total and directional generalized differentiability

We extend the interval and fuzzy gH-differentiability to consider interval and fuzzy valued functions of several variables and to include directional gH-differentiability; the proposed setting is more general than the existing definitions in the literature and allows a unified view of total and direction gH-differentiability and for the computation of partial gH-derivatives, directional gH-derivative and level wise gH-differentiability in the fuzzy valued case. A concept of gH-differential is then deduced and its values are used to define an (abstract) local tangency property for a gH-differentiable function, similar to the well known tangency between a differentiable function and its tangent plane. The proposed new setting allows an analysis of conditions for local optimality (dominance with respect to interval and level-wise partial orders well known in the literature) in terms of directional gH-derivatives, including concepts of local convexity, and to formulate KKT-like conditions for non-dominated solutions in constrained optimization problems.