  We propose a class of multipliers correction methods to minimize a
differentiable function over the Stiefel manifold. The proposed methods combine
a function value reduction step with a proximal correction step. The former one
searches along an arbitrary descent direction in the Euclidean space instead of
a vector in the tangent space of the Stiefel manifold. Meanwhile, the latter
one minimizes a first-order proximal approximation of the objective function in
the range space of the current iterate to make Lagrangian multipliers
associated with orthogonality constraints symmetric at any accumulation point.
The global convergence has been established for the proposed methods.
Preliminary numerical experiments demonstrate that the new methods
significantly outperform other state-of-the-art first-order approaches in
solving various kinds of testing problems.
