An Augmented Lagrangian Method for ₁-Regularized Optimization Problems with Orthogonality Constraints

A class of ₁-regularized optimization problems with orthogonality constraints has been used to model various applications arising from physics and information sciences, e.g., compressed modes for variational problems. Such optimization problems are difficult to solve due to the nonsmooth objective function and nonconvex constraints. Existing methods either are not applicable to such problems or lack convergence analysis, e.g., the splitting of orthogonality constraints (SOC) method. In this paper, we propose a proximal alternating minimized augmented Lagrangian method that hybridizes the augmented Lagrangian method and the proximal alternating minimization scheme. It is shown that the proposed method has the so-called subsequence convergence property; i.e., there exists at least one convergent subsequence, and any convergent subsequence converges to a Karush--Kuhn--Tucker point of an equivalent minimization problem. Experiments on the problem of compressed modes illustrate that the proposed method is noticeably faster than the SOC method.