Transforming into an exact penalty function model with convex compact constraints yields efficient infeasible approaches for optimization problems with orthogonality constraints. For smooth and L21-norm regularized cases, these infeasible approaches adopt simple and orthonormalization-free updating schemes and show high efficiency in some numerical experiments. However, to avoid orthonormalization while enforcing the feasibility of the final solution, these infeasible approaches introduce a quadratic penalty term, where an inappropriate penalty parameter can lead to numerical inefficiency. Inspired by penalty-free approaches for smooth optimization problems, we proposed a sequential linearized proximal gradient method (SLPG) for a class of optimization problems with orthogonality constraints and nonsmooth regularization term. This approach alternatively takes tangential steps and normal steps to improve the optimality and feasibility respectively. In SLPG, the orthonormalization process is invoked only once at the last step if high precision for feasibility is needed, showing that main iterations in SLPG are orthonormalization-free. Besides, both the tangential steps and normal steps do not involve the penalty parameter, and thus SLPG is penalty-free and avoids the inefficiency caused by possible inappropriate penalty parameter. We analyze the global convergence properties of SLPG where the tangential steps are inexactly computed. By inexactly computing tangential steps, for smooth cases and L21-norm regularized cases, SLPG has a closed-form updating scheme, which leads to cheap tangential steps. Numerical experiments illustrate the advantages of SLPG when compared with existing first-order methods.
View A Penalty-free Infeasible Approach for a Class of Nonsmooth Optimization Problems over the Stiefel Manifold