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.

翻译：本文提出了基于黎曼流形上谱尺度Broyden族的改进无记忆拟牛顿方法。该方法通过在流形上无记忆自尺度Broyden族的搜索方向中引入一个参数，并采用广义映射替代向量传输。这一思想已在黎曼共轭梯度法的通用框架中被提出，该框架允许使用向量传输、缩放向量传输或逆回缩。我们证明，在参数满足特定假设条件下，搜索方向满足充分下降条件。此外，在Wolfe条件下证明了所提方法的全局收敛性。通过数值实验与现有方法（包括黎曼共轭梯度法和无记忆谱尺度Broyden族）进行对比，结果表明采用BFGS公式的所提方法适用于求解斜流形上的非对角代价函数最小化问题。