Directed Subdifferentiable Functions and the Directed Subdifferential   without Delta-Convex Structure
  We show that the directed subdifferential introduced for differences of convex (delta-convex, DC) functions by Baier and Farkhi can be constructed from the directional derivative without using any information on the DC structure of the function. The new definition extends to a more general class of functions, which includes Lipschitz functions definable on o-minimal structure and quasidifferentiable functions. 
Robert Baier, Elza Farkhi and Vera Roshchina
math.OC