  This paper considers optimization problems on the Stiefel manifold
$X^{\mathsf{T}}X=I_p$, where $X\in \mathbb{R}^{n \times p}$ is the variable and
$I_p$ is the $p$-by-$p$ identity matrix. A framework of constraint preserving
update schemes is proposed by decomposing each feasible point into the range
space of $X$ and the null space of $X^{\mathsf{T}}$. While this general
framework can unify many existing schemes, a new update scheme with low
complexity cost is also discovered. Then we study a feasible
Barzilai-Borwein-like method under the new update scheme. The global
convergence of the method is established with an adaptive nonmonotone line
search. The numerical tests on the nearest low-rank correlation matrix problem,
the Kohn-Sham total energy minimization and a specific problem from statistics
demonstrate the efficiency of the new method. In particular, the new method
performs remarkably well for the nearest low-rank correlation matrix problem in
terms of speed and solution quality and is considerably competitive with the
widely used SCF iteration for the Kohn-Sham total energy minimization.
