  The optimisation of nonsmooth, nonconvex functions without access to
gradients is a particularly challenging problem that is frequently encountered,
for example in model parameter optimisation problems. Bilevel optimisation of
parameters is a standard setting in areas such as variational regularisation
problems and supervised machine learning. We present efficient and robust
derivative-free methods called randomised Itoh--Abe methods. These are
generalisations of the Itoh--Abe discrete gradient method, a well-known scheme
from geometric integration, which has previously only been considered in the
smooth setting. We demonstrate that the method and its favourable energy
dissipation properties are well-defined in the nonsmooth setting. Furthermore,
we prove that whenever the objective function is locally Lipschitz continuous,
the iterates almost surely converge to a connected set of Clarke stationary
points. We present an implementation of the methods, and apply it to various
test problems. The numerical results indicate that the randomised Itoh--Abe
methods are superior to state-of-the-art derivative-free optimisation methods
in solving nonsmooth problems while remaining competitive in terms of
efficiency.
