ABSTRACT
As we have seen in Chapter 1, differentiable convex functions and generalized convex functions can be characterized by means of the gradient function. However, when the function fails to be differentiable, the convexity may be given in terms of a generalized derivative (if it exists). For instance, if f : Rn → R is directionally differentiable, then f is convex if and only if
f ′(x; y − x) ≤ f(y)− f(x), for all x, y ∈ Rn. Various types of generalized derivatives exist in literature and some of
them have been studied in Chapter 2. Most of the generalized derivatives, like the directional derivative, the Dini derivative, and the Clarke derivative, share a very important property, namely, positive homogeneity as a function of the direction. Motivated by this fact an attempt was made to unify the generalized derivatives by considering a bifunction h(x; d) with values in R∪{±∞}, where x refers to a point in the domain of the function and d is a direction in Rn. Komlo´si [132] defined generalized convexity in terms of this bifunction h(x; d). This definition encompassed most of the existing definitions involving the generalized derivatives.
