Are Generalized Derivatives Sseful for Generalized Convex Functions?

We present a review of some ad hoc subdifferentials which have been devised for the needs of generalized convexity such as the quasi-subdifferentials of Greenberg-Pierskalla, the tangential of Crouzeix, the lower subdifferential of Plastria, the infradifferential of Gutiérrez, the subdifferentials of Martínez-Legaz-Sach, Penot-Volle, Thach. We complete this list by some new proposals. We compare these specific subdifferentials to some all-purpose subdifferentials used in nonsmooth analysis. We give some hints about their uses. We also point out links with duality theories.