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Riemannian conjugate gradient methods with inverse retraction

Author

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  • Xiaojing Zhu

    (Shanghai University of Electric Power)

  • Hiroyuki Sato

    (Kyoto University)

Abstract

We propose a new class of Riemannian conjugate gradient (CG) methods, in which inverse retraction is used instead of vector transport for search direction construction. In existing methods, differentiated retraction is often used for vector transport to move the previous search direction to the current tangent space. However, a different perspective is adopted here, motivated by the fact that inverse retraction directly measures the displacement from the current to the previous points in terms of tangent vectors at the current point. The proposed algorithm is implemented with the Fletcher–Reeves and the Dai–Yuan formulae, respectively, and global convergence is established using modifications of the Riemannian Wolfe conditions. Computational details of the practical inverse retractions over the Stiefel and fixed-rank manifolds are discussed. Numerical results obtained for the Brockett cost function minimization problem, the joint diagonalization problem, and the low-rank matrix completion problem demonstrate the potential effectiveness of Riemannian CG with inverse retraction.

Suggested Citation

  • Xiaojing Zhu & Hiroyuki Sato, 2020. "Riemannian conjugate gradient methods with inverse retraction," Computational Optimization and Applications, Springer, vol. 77(3), pages 779-810, December.
    • Handle: RePEc:spr:coopap:v:77:y:2020:i:3:d:10.1007_s10589-020-00219-6
    DOI: 10.1007/s10589-020-00219-6
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    References listed on IDEAS

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    1. P.-A. Absil & I. Oseledets, 2015. "Low-rank retractions: a survey and new results," Computational Optimization and Applications, Springer, vol. 62(1), pages 5-29, September.
    2. Wenyu Sun & Ya-Xiang Yuan, 2006. "Optimization Theory and Methods," Springer Optimization and Its Applications, Springer, number 978-0-387-24976-6, December.
    3. Hiroyuki Sato, 2016. "A Dai–Yuan-type Riemannian conjugate gradient method with the weak Wolfe conditions," Computational Optimization and Applications, Springer, vol. 64(1), pages 101-118, May.
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    Cited by:

    1. Yasushi Narushima & Shummin Nakayama & Masashi Takemura & Hiroshi Yabe, 2023. "Memoryless Quasi-Newton Methods Based on the Spectral-Scaling Broyden Family for Riemannian Optimization," Journal of Optimization Theory and Applications, Springer, vol. 197(2), pages 639-664, May.
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    3. Yuya Yamakawa & Hiroyuki Sato, 2022. "Sequential optimality conditions for nonlinear optimization on Riemannian manifolds and a globally convergent augmented Lagrangian method," Computational Optimization and Applications, Springer, vol. 81(2), pages 397-421, March.
    4. Hiroyuki Sato, 2023. "Riemannian optimization on unit sphere with p-norm and its applications," Computational Optimization and Applications, Springer, vol. 85(3), pages 897-935, July.

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