Generalized global monotonicity of generalized derivatives

The gradient and the several kinds of its generalizations provide a very efficient tool in characterizing important properties of functions. Convexity and generalized convexity, which are central properties in many branches of Operational Research, can also be characterized by special properties (monotonicity and generalized monotonicity) of the gradient map in the smooth case and by that of the Dini derivatives in the nonsmooth case. It is shown in this paper how quasiconvexity, pseudoconvexity and (strict) pseudoconvexity of lower semicontinuous functions can be characterized via quasimonotonicity and (strict) pseudomonotonicity of different types of generalized derivatives, including the Dini, Dini-Hadamard, Clarke and Rockafellar derivatives.