Modified Memoryless Spectral-Scaling Broyden Family on Riemannian Manifolds
本文在黎曼流形上提出一种改进的无记忆拟牛顿方法，通过向搜索方向添加参数并使用一般映射替代向量传输，证明了充分下降条件和全局收敛性，数值实验表明该方法在斜流形上的非对角代价函数最小化问题中表现良好。
Abstract This paper presents modified memoryless quasi-Newton methods based on the spectral-scaling Broyden family on Riemannian manifolds. The method involves adding one parameter to the search direction of the memoryless self-scaling Broyden family on the manifold. Moreover, it uses a general map instead of vector transport. This idea has already been proposed within a general framework of Riemannian conjugate gradient methods where one can use vector transport, scaled vector transport, or an inverse retraction. We show that the search direction satisfies the sufficient descent condition under some assumptions on the parameters. In addition, we show global convergence of the proposed method under the Wolfe conditions. We numerically compare it with existing methods, including Riemannian conjugate gradient methods and the memoryless spectral-scaling Broyden family. The numerical results indicate that the proposed method with the BFGS formula is suitable for solving an off-diagonal cost function minimization problem on an oblique manifold.