The Directed Subdifferential of DC and Quasidifferentiable Functions

Elza Farkhi, Robert Baier, Vera Roshchina

Following an idea of A. Rubinov, the directed subdifferential of a difference of convex (DC) functions is defined as difference of the the two convex subdifferentials embedded in the space of directed sets. This approach is later applied to define directed subdifferential of
quasidiffererentiable (QD) functions which have DC directional derivatives.
Seven of the eight axioms of A. Ioffe for a subdifferential hold for the directed subdifferential, among them the exact sum rule (satisfied as equality).
While preserving the most important properties of the quasidifferential, such as exact
calculus rules, the directed subdifferential lacks the major drawbacks of quasidifferential:
non-uniqueness and ‘‘inflation in size’’ of the two convex sets representing the quasidifferential after applying calculus rules.
The visualization of the directed subdifferential is called the
Rubinov subdifferential. The latter contains the Dini-Hadamard subdifferential
as its convex part, the Dini-Hadamard superdifferential as its concave part,
and its convex hull equals the Michel-Penot subdifferential. Hence, in general,
the Rubinov subdifferential contains less critical points than the Michel-Penot
subdifferential, while the sharp necessary and sufficient optimality conditions
in terms of the Dini-Hadamard subdifferential are recovered by the convex part
of the directed subdifferential.
The directed subdifferential allows easily to detect and distinguish candidates for
a maximum and for a minimum, as well as ascent and descent directions
from its visualization.