Automatic directional differentiation of nonsmooth composite functions

We consider continuous functions that are defined by programs for their evaluation. The basic arithmetic operations and univariate special functions are real analytic in the interior of their domains. However, another frequent ingredient, the absolute value function, has a kink at the origin, which occurs similarly for max and min. A slightly more serious complication arises with the introduction of Euclidean vector norms. It is shown here that the resulting class of composite functions is still directionally real analytic and we develop formulas for propagating the corresponding directional Taylor-coefficients in the forward mode of automatic differentiation. Finally, we discuss possibilities for using the reverse mode to compute generalized gradients.