A First-Order Primal-Dual Method for Nonconvex Constrained Optimization Based On the Augmented Lagrangian

Abstract: Nonlinearly constrained nonconvex and nonsmooth optimization models play an increasingly important role in machine learning, statistics and data analytics. In this paper, based on the augmented Lagrangian function we introduce a flexible first-order primal-dual method, to be called nonconvex auxiliary problem principle of augmented Lagrangian (NAPP-AL), for solving a class of nonlinearly constrained nonconvex and nonsmooth optimization problems. We demonstrate that NAPP-AL converges to a stationary solution at the rate of o(1/\sqrt{k}), where k is the number of iterations. Moreover, under an additional error bound condition (to be called VP-EB in the paper), we further show that the convergence rate is in fact linear. Finally, we show that the famous Kurdyka- Lojasiewicz property and the metric subregularity imply the afore-mentioned VP-EB condition.