ABSTRACT
A well-known aspect of a differentiable convex function is that it is closely related to monotonicity. It is known that a differentiable function is convex if and only if its gradient is a monotone map. Generalized monotone maps also provide a characterization for generalized convex functions and they are not new in the literature. For instance, a differentiable function is quasiconvex if and only if its gradient has a property that, quite naturally, was called quasimonotonicity. It is very interesting to observe that the first appearance of generalized monotonicity (well before the birth of this terminology) dates back to 1936. Even more remarkable, it occurred independently and almost at the same time in two seminal articles: the one, by Georgescu-Roegen (1936), dealt with the concept of a local preference in consumer theory of economics, and the other one, by Wald (1936), contained the first rigorous proof of the existence of a competitive general equilibrium. It is also remarkable that various axioms on revealed preferences in consumer theory are in fact generalized monotone conditions. The recognition of close links between the already well-established field of generalized convexity and the relatively undeveloped field of generalized monotonicity gave a boost to both and led to a major increase in research activities. Today generalized monotonicity is frequently used in complementarity problems, variational inequalities, fixed point theory, optimization, equilibrium problems, and so on. This can be extended to nondifferentiable functions as well, through the use of generalized directional derivatives, subdifferentials, or multivalued maps. Similar connections were discovered between generalized convex functions and certain classes of maps, generically called generalized monotone.
