Abstract
We prove the convergence of the proximal point algorithm for finding the unique minimizer of a strongly quasiconvex function in general nonlinear Hadamard spaces, generalizing a recent result due to F. Lara. Our argument is rather elementary and brief and relies only on a few properties of strongly quasiconvex functions and their proximal operators which are established here for the first time over these nonlinear spaces. In particular, our convergence proof is fully effective and actually yields fast (ranging up to linear) rates of convergence for the iterates towards the solution and for the function values towards the minimum. These rates are novel even in the context of Euclidean spaces.
| Original language | English |
|---|---|
| Pages (from-to) | 1438-1453 |
| Number of pages | 16 |
| Journal | Optimization Methods and Software |
| Volume | 40 |
| Issue number | 6 |
| Early online date | 29 Aug 2025 |
| DOIs | |
| Publication status | Published - 29 Aug 2025 |
Bibliographical note
Publisher Copyright:© 2025 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
Acknowledgements
This work benefited from conversations with Sorin-Mihai Grad, Laurenţiu Leuştean, Nicoleta Dumitru and Ulrich Kohlenbach. I want to thank the anonymous referees for many very helpful comments and in particular some very interesting references, the inclusion of which improved the manuscript.Funding
This work was supported by the ‘Deutsche Forschungsgemeinschaft’ Project DFG KO 1737/6-2.
Keywords
- Proximal point algorithm
- rates of convergence
- strongly quasiconvex functions
- Hadamard spaces
- proof mining
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