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

Modified Memoryless Spectral-Scaling Broyden Family on Riemannian Manifolds

“…The idea of behind the memoryless spectral-scaling Broyden family is very simple: replace Hk−1 with id T x k−1 M . In [16], a memoryless spectral-scaling Broyden family on a Riemannian manifold is proposed by replacing Hk−1 with id T x k M . To guarantee global convergence, they replaced y k−1 by z k−1 ∈ T x k M satisfying the following conditions [16, (27)]: for positive constants ν, ν > 0,…”

“…In [19], Ring and Wirth proposed the BFGS method, which has a global convergence property under some convexity assumptions. Narushima et al [16] proposed memoryless quasi-Newton methods based on the spectral-scaling Broyden family on Riemannian manifolds. They extended the memoryless spectral-scaling Broyden family in Euclidean space to Riemannian manifolds with an additional modification to ensure a sufficient descent condition.…”

“…Therefore, it is worth extending it to Riemannian manifolds. Our method is based on the memoryless quasi-Newton methods on Riemannian manifolds proposed by Narushima et al [16] as well as on the modification by Kou and Dai [9]. It uses a general map to transport vectors similarly to the general framework of Riemannian conjugate gradient methods [25].…”

“…Moreover, we present global convergence analyses under the Wolfe conditions (see Theorem 4.2). Furthermore, we describe the results of numerical experiments comparing our method with the existing ones, including Riemannian conjugate gradient methods [20] and the memoryless spectral-scaling Broyden family on Riemannian manifolds [16]. The key advantages of the proposed methods are the added parameter ξ k−1 and support for maps other than vector transports.…”

“…As shown in the numerical experiments, the proposed method may outperform the existing methods depending on how the parameter ξ k−1 is chosen. It has an advantage over [16] in that it can use a map such as an inverse retraction, which is not applicable in [16].…”