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Peer-reviewedCC BY 4.0Mathematics

An Improved Riemannian Trust-Region Method Combining Line Search Techniques with Applications in Matrix Optimization

  • Jiahui Yuan1
  • Tao Yan1
  1. 1School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China
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Abstract

We propose a hybrid Riemannian optimization framework that combines trust-region methods with retraction-based backtracking line search to reduce the cost of repeatedly solving subproblems corresponding to rejected trial steps. If a trust-region step is not accepted, the algorithm reuses the descent direction and performs a backtracking line search instead of resolving the subproblem. Under the stated assumptions, we establish lim infk→∞∥gradf(xk)∥xk=0. Numerical experiments on three matrix optimization problems with orthogonality constraints, namely Brockett cost function minimization on the Stiefel manifold, Rayleigh quotient minimization on the sphere manifold, and independent component analysis on the Stiefel manifold, demonstrate that RTR-LS and LRTR-LS are computationally more efficient than classical Riemannian trust-region methods.

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Cite

BibTeXRIS
BibTeX
@article{yuan2026improved,
  title = {{An Improved Riemannian Trust-Region Method Combining Line Search Techniques with Applications in Matrix Optimization}},
  author = {Jiahui Yuan and Tao Yan},
  journal = {Mathematics},
  year = {2026},
  month = nov,
  doi = {10.3390/math14214098},
  url = {https://www.mdpi.com/2227-7390/14/21/4098},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - An Improved Riemannian Trust-Region Method Combining Line Search Techniques with Applications in Matrix Optimization
AU  - Jiahui Yuan
AU  - Tao Yan
PY  - 2026
DA  - 2026/11/08
JO  - Mathematics
DO  - 10.3390/math14214098
UR  - https://www.mdpi.com/2227-7390/14/21/4098
AB  - We propose a hybrid Riemannian optimization framework that combines trust-region methods with retraction-based backtracking line search to reduce the cost of repeatedly solving subproblems corresponding to rejected trial steps. If a trust-region step is not accepted, the algorithm reuses the descent direction and performs a backtracking line search instead of resolving the subproblem. Under the stated assumptions, we establish lim infk→∞∥gradf(xk)∥xk=0. Numerical experiments on three matrix optimization problems with orthogonality constraints, namely Brockett cost function minimization on the Stiefel manifold, Rayleigh quotient minimization on the sphere manifold, and independent component analysis on the Stiefel manifold, demonstrate that RTR-LS and LRTR-LS are computationally more efficient than classical Riemannian trust-region methods.
SN  - 2227-7390
ER  -