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

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.