  We present a reformulation of optimization problems over the Stiefel manifold
by using a Cayley-type transform, named the generalized left-localized Cayley
transform, for the Stiefel manifold. The reformulated optimization problem is
defined over a vector space, whereby we can apply directly powerful
computational arts designed for optimization over a vector space. The proposed
Cayley-type transform enjoys several key properties which are useful to (i)
study relations between the original problem and the proposed problem; (ii)
check the conditions to guarantee the global convergence of optimization
algorithms. Numerical experiments demonstrate that the proposed algorithm
outperforms the standard algorithms designed with a retraction on the Stiefel
manifold.
