Multipliers Correction Methods for Optimization Problems over the Stiefel Manifold (CSIAM Transactions on Applied Mathematics, 2021)
We propose a class of multipliers correction methods to minimize a differentiable function over the Stiefel manifold. The proposed methods combine a functionvalue reduction step with a proximal correction step. The former one searches alongan arbitrary descent direction in the Euclidean space instead of a vector in the tangentspace of the Stiefel manifold. Meanwhile, the latter one minimizes a first-order proximal approximation of the objective function in the range space of the current iterate tomake Lagrangian multipliers associated with orthogonality constraints symmetric atany accumulation point. The global convergence has been established for the proposedmethods. Preliminary numerical experiments demonstrate that the new methods significantly outperform other state-of-the-art first-order approaches in solving variouskinds of testing problems.
Keywords: Multipliers correction; proximal approximation; orthogonality constraint; Stiefel manifold.