WORST-CASE COMPLEXITY OF SMOOTHING QUADRATIC REGULARIZATION METHODS FOR NON-LIPSCHITZIAN OPTIMIZATION

In this paper, we propose a smoothing quadratic regularization (SQR) algorithm for solving a class of nonsmooth nonconvex, perhaps even non-Lipschitzian minimization problems, which has wide applications in statistics and sparse reconstruction. The proposed SQR algorithm is a first order method. At each iteration, the SQR algorithm solves a strongly convex quadratic minimization problem with a diagonal Hessian matrix, which has a simple closed-form solution that is inexpensive to calculate. We show that the worst-case complexity of reaching an is an element of scaled stationary point is O(is an element of(-2)). Moreover, if the objective function is locally Lipschitz continuous, the SQR algorithm with a slightly modified updating scheme for the smoothing parameter and iterate can obtain an is an element of Clarke stationary point in at most O(is an element of(-3)) iterations.