When studying optimization problems, the mathematician has to make use of certain tools. In the smooth case, for example, one such tool is the
derivative; in the nonsmooth case, the derivative concept is naturally replaced
by the idea of a directional derivative or one of its generalizations. But there
is a catch: any such generalization must be consistent with simpler cases,
e.g., if the function is directionally differentiable, then the generalized
directional derivative should be the same as the ordinary directional deriv-
ative. Otherwise the generalization-even though aesthetically elegant-may
have little or no relevance to the optimization problem.