  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 over the Stiefel manifold. We propose a framework for
developing subgradient-based methods and establish 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.
