A Feasible Augmented Lagrangian Method for Non-lipschitz Nonconvex Programming

We consider a class of constrained optimization problems where the objective function is a sum of a smooth function and a nonconvex non-Lipschitz function. Many problems in sparse portfolio selection, edge preserving image restoration and signal processing can be modelled in this form. First we propose the concept of the Karush-Kuhn-Tucker (KKT) stationary condition for the non-Lipschitz problem and show that it is necessary for optimality under a constraint qualification called the relaxed constant rank linear dependence (RCPLD) condition which is weaker than the Mangasarian-Fromovitz constraint qualification and holds automatically if all the constraint functions are linear. Then we propose a feasible augmented Lagrangian method (FAL) in which the augmented Lagrangian subproblems is solved by a non-monotone proximal gradient method. In contrast to the classical augmented Lagrangian method which may converge to an infeasible point, our method is guaranteed to converge to a feasible point. Moreover, under the RCPLD condition we show that any accumulation point of the sequence generated by our method is a KKT point of the original problem. Finally we conduct numerical experiments to compare the performance of our FAL method and the interior point (IP) method for solving two sparse portfolio selection models. The numerical results demonstrate that our method is not only comparable to the IP method in terms of solution quality, but also substantially faster than the IP method.