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 [Formula: see text], where k is the number of iterations. Moreover, under an additional error bound condition (to be called HVP-EB in the paper) with exponent [Formula: see text], we further show the global convergence of NAPP-AL. Additionally, if [Formula: see text], then we furthermore show that the convergence rate is in fact linear. Finally, we show that the well-known Kurdyka-Łojasiewicz property and the Hölderian metric subregularity imply the aforementioned HVP-EB condition. We demonstrate that under mild conditions, NAPP-AL can also be interpreted as a variant of the forward-backward operator splitting method in this context. Funding: This work was supported by the National Natural Science Foundation of China [Grant 71871140]. more...