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.

The hybrid procedure is an efficient technique for improving the global and numerical performance of iterative algorithms for large-scale monotone nonlinear problems. This is achieved by integrating two or more methods into a unified framework. In this study, we present a hybrid of the double step-length method and the Picard-Mann iterative technique for solving monotone nonlinear equations with convex constraints. By combining the Picard-Mann approach with a newly proposed scheme, we obtain an iterative method that reduces computational cost and achieves faster convergence. The acceleration parameter is determined by evaluating the difference between the Broyden update and its approximation using the Frobenius norm. The global convergence of the proposed method is proved, and a Q-linear convergence rate is also established. Numerical experiments demonstrate that the proposed approach is computationally efficient for solving large-scale nonlinear equations compared with existing methods. Finally, the method is applied to signal processing and image restoration problems, highlighting its practical relevance.