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We present a reformulation of optimization problems over the Stiefel manifoldby using a Cayley-type transform, named the generalized left-localized Cayleytransform, for the Stiefel manifold. The reformulated optimization problem isdefined over a vector space, whereby we can apply directly powerfulcomputational arts designed for optimization over a vector space. The proposedCayley-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 optimizationalgorithms. Numerical experiments demonstrate that the proposed algorithmoutperforms the standard algorithms designed with a retraction on the Stiefelmanifold.

Generalized Left-Localized Cayley Parametrization for Optimization with Orthogonality Constraints