Nonconvex and Nonsmooth Optimization with Generalized Orthogonality Constraints: An Approximate Augmented Lagrangian Method

Nonconvex and nonsmooth optimization problems with linear equation and generalized orthogonality constraints have wide applications. These problems are difficult to solve due to nonsmooth objective function and nonconvex constraints. In this paper, by introducing an extended proximal alternating linearized minimization (EPALM) method, we propose a framework based on the augmented Lagrangian scheme (EPALMAL). We also show that the EPALMAL method has global convergence in the sense that every bounded sequence generated by the EPALMAL method has at least one convergent subsequence that converges to the Karush–Kuhn–Tucker point of the original problem. Experiments on a variety of applications, including compressed modes and multivariate data analysis, have demonstrated that the proposed method is noticeably efficient and achieves comparable performance with existing methods.