  This paper is devoted to studying the first-order variational analysis of
non-convex and non-differentiable functions that may not be subdifferentially
regular. To achieve this goal, we entirely rely on two concepts of directional
derivatives known as subderivative and semi-derivative. We establish the exact
chain and sum rules for this class of functions via these directional
derivatives. These calculus rules provide an implementable auto-differentiation
process such as back-propagation in composite functions. The latter calculus
rules can be used to identify the directional stationary points defined by the
subderivative. We show that the distance function of a geometrically derivable
constraint set is semi-differentiable, which opens the door for designing
first-order algorithms for non-Clarke regular constrained optimization
problems. We propose a first-order algorithm to find a directional stationary
point of non-Clarke regular and perhaps non-Lipschitz functions. We introduce a
descent property under which we establish the non-asymptotic convergence of our
method with rate $O(\varepsilon^{-2})$, akin to gradient descent for smooth
minimization. We show that the latter descent property holds for free in some
interesting non-amenable composite functions, in particular, it holds for the
Moreau envelope of any bounded-below function.
