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.

翻译：正交约束二次极小化问题（QMPO）在科学与工程的许多应用中扮演着重要角色。然而，现有的一些方法在处理大规模QMPO时可能面临精度低或计算量大的问题。Krylov子空间方法在大规模优化问题中广受欢迎。本文提出了一种用于求解大规模QMPO的块Lanczos方法。在所提方法中，原始问题被投影成一个小型问题，并采用黎曼信赖域方法求解约化后的QMPO。建立了关于最优解、最优目标函数值、乘子以及KKT误差的收敛性结果。此外，我们给出了最优解的收敛速度，并证明若块Lanczos过程终止，则得到精确的KKT解。数值实验展示了所提算法的数值表现，并证明其优于许多用于大规模正交约束二次极小化问题的先进算法。