A Geometric Integration Approach to Nonsmooth, Nonconvex Optimisation

Erlend S. Riis, Matthias J. Ehrhardt, G. R. W. Quispel, Carola Bibiane Schönlieb

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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 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 can be superior to state-of-the-art derivative-free optimisation methods in solving nonsmooth problems while still remaining competitive in terms of efficiency.

Original languageEnglish
Pages (from-to)1351–1394
JournalFoundations of Computational Mathematics
Volume22
Early online date29 Jul 2021
DOIs
Publication statusPublished - 31 Oct 2022

Keywords

  • Bilevel optimisation
  • Clarke subdifferential
  • Derivative-free optimisation
  • Discrete gradient methods
  • Geometric numerical integration
  • Nonconvex optimisation
  • Nonsmooth optimisation

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics

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