Proximal bundle methods are among the most successful approaches for convex and nonconvex optimization problems in linear spaces and it is natural to extend these methods to the manifold setting. In this paper we propose a proximal bundle method for solving nonsmooth, nonconvex optimization problems on Riemannian manifolds. At every iteration, by using the proximal bundle method a candidate descent direction is computed and by employing a restricted-step procedure and a retraction the next iterate is built. The global convergence, starting from any point, is proved in the sense that if the number of serious iterates is finite then the last serious iterate is stationary and otherwise every accumulation point of the serious iterate sequence is stationary. At the end, some numerical experiments are provided to illustrate the effectiveness of the proposed method and clarify the theoretical results.

中文翻译：

---

一种用于黎曼流形上非光滑优化的近端束算法

近端束方法是线性空间中凸和非凸优化问题最成功的方法之一，很自然地将这些方法扩展到流形设置。在本文中，我们提出了一种近端束方法来解决黎曼流形上的非光滑、非凸优化问题。在每次迭代中，通过使用近端束方法计算候选下降方向，并通过采用受限步长过程和回退来构建下一次迭代。从任何点开始的全局收敛性在以下意义上被证明：如果严重迭代的数量是有限的，则最后一个严重迭代是平稳的，否则严重迭代序列的每个累积点都是平稳的。在末尾，