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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.

A framework of constraint preserving update schemes for optimization on Stiefel manifold