On the Subdifferentials of Quasiconvex and Pseudoconvex Functions and Cyclic Monotonicity*

The notions of cyclic quasimonotonicity and cyclic pseudomonotonicity are introduced. A classical result of convex analysis concerning the cyclic monotonicity of the Fenchel]Moreau.subdifferential of a convex function is extended to corresponding results for the Clarke]Rockafellar subdifferential of quasiconvex and pseudoconvex functions. The notion of proper quasimonotonicity is also introduced. It is shown that this new notion retains the characteristic property of quasimonotonicity i.e., a lower semicontinuous function is quasiconvex if and only if its Clarke]Rockafellar subdifferential is properly quasimonotone., while it is also related to the KKM property of multivalued maps; this makes it more useful in applications to variational inequalities