## A framework of constraint preserving update schemes for optimization on Stiefel manifold.(English)Zbl 1325.49037

Summary: 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.

### MSC:

 49M37 Numerical methods based on nonlinear programming 49Q99 Manifolds and measure-geometric topics 65K05 Numerical mathematical programming methods 90C30 Nonlinear programming

### Software:

SparseMatrix; KSSOLV
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