  We study the minimization of a quadratic over Stiefel manifolds (the set of
all orthogonal $r$-frames in \IR^n), which has applications in high-dimensional
semi-supervised classification tasks. To reduce the computational complexity,
sequential subspace methods(SSM) are employed to convert the high-dimensional
minimization problems to low-dimensional ones. In this paper, we are interested
in attaining an optimal solution of good quality, i.e., a ``qualified" critical
point. Qualified critical points are those critical points, at which the
associated multiplier matrix meets some upper bound condition. These critical
points enjoy the global optimality in special quadratic problems. For a general
quadratic,
  SSM computes a sequence of ``qualified critical points" in its
low-dimensional ``surrogate regularized models". The convergence to a qualified
critical point is ensured, whenever each SSM subspace is constructed by the
following vectors: (i) a set of orthogonal unit vectors associated with the
current iterate, (ii) a set of vectors corresponding to the gradient of the
objective, and (iii) a set of eigenvectors associated with the smallest $r$
eigenvalues of the system matrix. In addition, when Newton direction vectors
are included in subspaces, the convergence of SSM can be accelerated
significantly.
