On generalized cyclically monotone operators and proper quasimonotonictty

Abstract

In this paper we consider an abstract subdifferential that fulfills a prioria weak type of a mean value property. We survey and extend some recent results connecting the gener-alized convexity of nonsmooth functions with the generalized cyclic monotonidty of their subdifferentials. It is shown that, for a large class of subdifferentials, a Isc function is quasiconvex if and only if its subdifferential is a cyclically quasimonotone operator. An analogous property holds for pseudoconvexity. It is also shown that the subdiffer-ential of a quasiconvex function is properly quasimonotone. This property is slightly stronger than quasimonotonicity, and is more useful in applications connected with variational inequalities

Keywords:

• Generalized Convexity
• Generalized Monotonicity
• Cyclically Monotone Opera-Tors
• Abstract Subdifferential