Memoryless Quasi-Newton Methods Based on the Spectral-Scaling Broyden Family for Riemannian Optimization

We consider iterative methods for unconstrained optimization on Riemannian manifolds. Though memoryless quasi-Newton methods are effective for large-scale unconstrained optimization in the Euclidean space, they have not been studied over Riemannian manifolds. Therefore, in this paper, we propose a memoryless quasi-Newton method in Riemannian manifolds. The proposed method is based on the spectral-scaling Broyden family with additional modifications to ensure the sufficient descent condition. We present an algorithm for the proposed method that uses the Wolfe line search conditions and show that this algorithm guarantees global convergence. We emphasize that global convergence is guaranteed without any assumptions regarding the convexity of the objective function or the isometric property of the vector transport. In addition, we derive appropriate selections for the parameter vector contained in the proposed method. Numerical experiments are conducted to compare the proposed method with conventional conjugate gradient methods using typical test problems. The results show that the proposed method is superior to the tested conjugate gradient methods.