  Quadratic minimization problems with orthogonality constraints (QMPO) play an
important role in many applications of science and engineering. However, some
existing methods may suffer from low accuracy or heavy workload for large-scale
QMPO. Krylov subspace methods are popular for large-scale optimization
problems. In this work, we propose a block Lanczos method for solving the
large-scale QMPO. In the proposed method, the original problem is projected
into a small-sized one, and the Riemannian Trust-Region method is employed to
solve the reduced QMPO. Convergence results on the optimal solution, the
optimal objective function value, the multiplier and the KKT error are
established. Moreover, we give the convergence speed of optimal solution, and
show that if the block Lanczos process terminates, then an exact KKT solution
is derived. Numerical experiments illustrate the numerical behavior of the
proposed algorithm, and demonstrate that it is more powerful than many
state-of-the-art algorithms for large-scale quadratic minimization problems
with orthogonality constraints.
