SIAM Review, Volume 66, Issue 2, Page 319-352, May 2024.
We consider optimization problems over the Stiefel manifold whose objective function is the summation of a smooth function and a nonsmooth function. Existing methods for solving this class of problems converge slowly in practice, involve subproblems that can be as difficult as the original problem, or lack rigorous convergence guarantees. In this paper, we propose a manifold proximal gradient method (ManPG) for solving this class of problems. We prove that the proposed method converges globally to a stationary point and establish its iteration complexity for obtaining an $\epsilon$-stationary point. Furthermore, we present numerical results on the sparse PCA and compressed modes problems to demonstrate the advantages of the proposed method. We also discuss some recent advances related to ManPG for Riemannian optimization with nonsmooth objective functions.

中文翻译：

---

Stiefel 流形及其他流形上的非平滑优化：近端梯度法和最新变体


《SIAM 评论》，第 66 卷，第 2 期，第 319-352 页，2024 年 5 月。

我们考虑 Stiefel 流形上的优化问题，其目标函数是平滑函数和非平滑函数的总和。解决此类问题的现有方法在实践中收敛缓慢，涉及的子问题可能与原始问题一样困难，或者缺乏严格的收敛保证。在本文中，我们提出了一种流形近端梯度法（ManPG）来解决此类问题。我们证明了所提出的方法全局收敛到一个驻点，并建立了其迭代复杂度以获得 $\epsilon$-驻点。此外，我们还给出了稀疏 PCA 和压缩模式问题的数值结果，以证明所提出方法的优点。我们还讨论了与 ManPG 有关的一些最新进展，用于非平滑目标函数的黎曼优化。