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Nonlinearly constrained nonconvex and nonsmooth optimization models play anincreasingly important role in machine learning, statistics and data analytics.In this paper, based on the augmented Lagrangian function we introduce aflexible first-order primal-dual method, to be called nonconvex auxiliaryproblem principle of augmented Lagrangian (NAPP-AL), for solving a class ofnonlinearly constrained nonconvex and nonsmooth optimization problems. Wedemonstrate that NAPP-AL converges to a stationary solution at the rate ofo(1/\sqrt{k}), where k is the number of iterations. Moreover, under anadditional error bound condition (to be called VP-EB in the paper), we furthershow that the convergence rate is in fact linear. Finally, we show that thefamous Kurdyka- Lojasiewicz property and the metric subregularity imply theafore-mentioned VP-EB condition.

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