This paper focus 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 on the Stiefel manifold. We propose a framework for developing subgradient-based methods and establish their convergence properties based on prior works. Preliminary numerical experiments further highlight that the proposed constraint dissolving approach enables the efficient and direct implementations of various unconstrained solvers to nonsmooth optimization problems over the Stiefel manifold.
View A Constraint Dissolving Approach for Nonsmooth Optimization over the Stiefel Manifold