This paper focuses on the minimization of a possibly nonsmooth objective function over the Stiefel manifold. The existing approaches either lack efficiency or can only tackle prox-friendly objective functions. We propose a constraint dissolving function named NCDF and show that it has the same first-order stationary points and local minimizers as the original problem in a neighborhood of the Stiefel manifold. Furthermore, we show that the Clarke subdifferential of NCDF is easy to achieve from the Clarke subdifferential of the objective function. Therefore, various existing approaches for unconstrained nonsmooth optimization can be directly applied to nonsmooth optimization problems over the Stiefel manifold. We propose a framework for developing subgradient-based methods and establishing their convergence properties based on prior works. Furthermore, based on our proposed framework, we can develop efficient approaches for optimization over the Stiefel manifold. Preliminary numerical experiments further highlight that the proposed constraint dissolving approach yields efficient and direct implementations of various unconstrained approaches to nonsmooth optimization problems over the Stiefel manifold.

中文翻译：

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Stiefel流形非光滑优化的约束消解方法

本文重点研究 Stiefel 流形上可能不光滑的目标函数的最小化。现有的方法要么缺乏效率，要么只能解决代理友好的目标函数。我们提出了一个名为 NCDF 的约束消解函数，并表明它与 Stiefel 流形邻域中的原始问题具有相同的一阶驻点和局部极小值。此外，我们还证明了 NCDF 的 Clarke 亚微分很容易从目标函数的 Clarke 亚微分得到。因此，各种现有的无约束非光滑优化方法可以直接应用于Stiefel流形上的非光滑优化问题。我们提出了一个框架，用于开发基于次梯度的方法并根据先前的工作建立其收敛特性。此外，基于我们提出的框架，我们可以开发有效的 Stiefel 流形优化方法。初步数值实验进一步强调，所提出的约束消解方法可以有效且直接地实现 Stiefel 流形上非光滑优化问题的各种无约束方法。