Two Globally Convergent Hybrid CG Methods with Adaptive Restart Strategy for Riemannian Optimization

Hybrid conjugate gradient methods (CGMs) with adaptive restart strategies have been well-researched in Euclidean space. In this paper, we extend two methods of this type to solve optimization problems on Riemannian manifolds. Firstly, we present two Riemannian hybrid CGMs, and their hybrid conjugate parameters are yielded by projection or convex combination of the classical parameters. The first is a Riemannian hybrid CGM that projects between the Dai-Yuan method and another flexible conjugate parameter. The second is a combination of projection and convex combination of the modified Riemannian Liu-Storey method and the modified Riemannian Hestenes-Stiefel method. In the framework of Riemannian CGMs, we apply a uniform adaptive restart strategy to both methods based on the hybrid conjugate parameters we proposed. The search direction of the presented methods satisfies the sufficient descent condition. Under the usual assumption and the weak Wolfe line search, we prove the global convergence of the two proposed methods. Finally, preliminary numerical results are reported and compared with several existing Riemannian CGMs, showing that our methods are effective.