Generalized global monotonicity of generalized derivatives
Sándor Komlósi · International Transactions in Operational Research · 1994
Abstract 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 (Karamardian and Schiable, 1990, JOTA , Vol. 66, pp. 37–46) and by that of the Dini derivatives in the nonsmooth case (Komlosim, 1991. Working paper, Janus Pannonius University; 1992a. Nonsmooth Optimization: Methods and Applications; Proceedings of the IVth International Workshop on Generalized Convexity , in press; Luc, 1991. Subgradients of quasiconvex functions). 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.