Abstract
In this paper, a Riemannian BFGS method is defined for minimizing a smooth function on a Riemannian manifold endowed with a retraction and a vector transport. The method is based on a Riemannian generalization of a cautious update and a weak line search condition. It is shown that, the Riemannian BFGS method converges (i) globally to a stationary point without assuming that the objective function is convex and (ii) superlinearly to a nondegenerate minimizer. The weak line search condition removes completely the need to consider the differentiated retraction. The joint diagonalization problem is used to demonstrate the performance of the algorithm with various parameters, line search conditions, and pairs of retraction and vector transport.
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Notes
- 1.
This mapping is not even required to be continuous in the definition. The smoothness is imposed in the convergence analyses.
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Acknowledgements
This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office. This work was supported by grant FNRS PDR T.0173.13.
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Huang, W., Absil, PA., Gallivan, K.A. (2016). A Riemannian BFGS Method for Nonconvex Optimization Problems. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-39929-4_60
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DOI: https://doi.org/10.1007/978-3-319-39929-4_60
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