An Augmented Lagrangian Method for ℓ1-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 non-smooth 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 (PAMAL) method that hybridizes the augmented Lagrangian method and the proximal alternating minimization scheme. It is shown that the proposed method has the so-called sub-sequence convergence property, i.e., there exists at least one convergent sub-sequence and any convergent sub-sequence converges to a Karush-Kuhn Tucker (KKT) 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.