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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\u2013Abe methods. These are generalisations of the Itoh\u2013Abe 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\u2013Abe methods can be superior to state-of-the-art derivative-free optimisation methods in solving nonsmooth problems while still remaining competitive in terms of efficiency.<\/jats:p>","DOI":"10.1007\/s10208-020-09489-2","type":"journal-article","created":{"date-parts":[[2021,7,29]],"date-time":"2021-07-29T22:02:26Z","timestamp":1627596146000},"page":"1351-1394","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["A Geometric Integration Approach to Nonsmooth, Nonconvex Optimisation"],"prefix":"10.1007","volume":"22","author":[{"given":"Erlend S.","family":"Riis","sequence":"first","affiliation":[]},{"given":"Matthias J.","family":"Ehrhardt","sequence":"additional","affiliation":[]},{"given":"G. R. 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