An Appropriate Subdifferential for Quasiconvex Functions

In this paper we introduce and study a subdifferential that is related to the quasi- convex functions, much as the Fenchel--Moreau subdifferential is related to the convex ones. It is defined for any lower semicontinuous function, through an appropriate combination of an abstract subdifferential and the normal cone to sublevel sets. We show that this "quasiconvex" subdifferential is always a cyclically quasimonotone operator that coincides with the Fenchel--Moreau subdifferential whenever the function is convex, and that under mild assumptions, the density of its domain in the domain of the function is equivalent to the quasiconvexity of the function. We also show that the "quasiconvex" subdifferential of a lower semicontinuous function contains the derivatives of its differentiable quasiaffine supports. As a consequence, it contains the subdifferential introduced by Martinez-Legaz and Sach in a recent paper [J. Convex Anal., 6 (1999), pp. 1--12]. Several other properties and calculus rules are also established.