A Hager–Zhang Riemannian conjugate gradient method for matrix approximation

We introduce a class of Hager–Zhang-type Riemannian conjugate gradient (CG) methods that generalize the framework of Sakai et al. (Applied Mathematics and Computation, 441 (2023) 127685) to arbitrary retractions while significantly extending its theoretical and practical scope. These methods ensure global convergence for non-convex problems without relying on strong convexity and inherently satisfy the sufficient descent property, independent of Riemannian line search conditions. A key algorithmic innovation is the introduction of an adaptive min(max) strategy to adjust the CG parameter ${\beta }_{k+1}$ using bounded parameters to ensure numerical stability. Furthermore, we extended classical Euclidean CG portfolio optimization to the sphere manifold, naturally enforcing budget constraints and improving robustness. Numerical experiments, available at GitHub repo, showed that our methods outperform classical Riemannian CG in iterations and computational time for large-scale problems.