Generalized Hukuhara Gâteaux and Fréchet derivatives of interval-valued functions and their application in optimization with interval-valued functions

In this article, the notions of gH-directional derivative, gH-Gâteaux derivative and gH-Fréchet derivative for interval-valued functions are proposed. The existence of gH-Fréchet derivative is shown to imply the existence of gH-Gâteaux derivative and the existence of gH-Gâteaux derivative is observed to indicate the presence of gH-directional derivative. For an interval-valued gH-Lipschitz function, it is proved that the existence of gH-Gâteaux derivative implies the existence of gH-Fréchet derivative. It is observed that for an interval-valued convex function on a linear space, the gH-directional derivative exists at any point for every direction. Concepts of linear and monotonic interval-valued functions are studied in the sequel. Further, it is shown that the proposed derivatives are useful to check the convexity of an interval-valued function and to characterize efficient points of an optimization problem with interval-valued objective function. It is observed that at an efficient point of an interval-valued function, none of its gH-directional derivatives dominates zero and the gH-Gâteaux derivative must contain zero. The entire study is supported by suitable illustrative examples.