Abstract 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 wellknown 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.
In REF , the application of a derivative-free discrete gradient method for nonconvex, nonsmooth functions is studied, and the method converges to a set of stationary points in the Clarke subdifferential framework.
A geometric integration approach to nonsmooth, nonconvex optimisation
arXiv: Optimization and Control
Mathematics