  An adaptive regularization algorithm using inexact function and derivatives
evaluations is proposed for the solution of composite nonsmooth nonconvex
optimization. It is shown that this algorithm needs at most
$O(|\log(\epsilon)|\,\epsilon^{-2})$ evaluations of the problem's functions and
their derivatives for finding an $\epsilon$-approximate first-order stationary
point. This complexity bound therefore generalizes that provided by [Bellavia,
Gurioli, Morini and Toint, 2018] for inexact methods for smooth nonconvex
problems, and is within a factor $|\log(\epsilon)|$ of the optimal bound known
for smooth and nonsmooth nonconvex minimization with exact evaluations. A
practically more restrictive variant of the algorithm with worst-case
complexity $O(|\log(\epsilon)|+\epsilon^{-2})$ is also presented.
