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 I(KM property of multivalued maps; this makes it more useful in applications to variational inequalities. (C) 1999 Academic Press.