A Framework of Constraint Preserving Update Schemes for Optimization on Stiefel Manifold

This paper considers optimization problems on the Stiefel manifold X^mathsfTX=Ip, where X∈ ℝn × p is the variable and Ip 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^mathsfT. 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.