We present a damped proximal augmented Lagrangian method (DPALM) for solving problems with a weakly-convex objective and convex linear/non-linear constraints. Instead of taking a full stepsize, DPALM adopts a (possibly) damped dual stepsize. We show that DPALM can produce a (near) \(\varepsilon \) -KKT point within \({\mathcal {O}}(\varepsilon ^{-2})\) outer iterations if each DPALM subproblem is solved to a proper accuracy. In addition, we establish overall oracle complexity (i.e., total number of evaluations of function values and (sub)gradients) of DPALM when the objective is either a regularized smooth function or in a regularized compositional form. For the former case, DPALM achieves a complexity of \(\widetilde{\mathcal {O}}\left( \varepsilon ^{-2.5} \right) \) to produce an \(\varepsilon \) -KKT point by applying an accelerated proximal gradient (APG) method to each DPALM subproblem. For the latter case, the complexity of DPALM is \(\widetilde{\mathcal {O}}\left( \varepsilon ^{-3} \right) \) to produce a near \(\varepsilon \) -KKT point by using an APG method to solve a Moreau-envelope smoothed version of each subproblem. Our outer iteration complexity and the overall complexity either generalize existing best ones from unconstrained or linear-constrained problems to convex-constrained ones, or improve over the best-known results on solving the same-structured problems. Furthermore, numerical experiments on linearly/quadratically constrained non-convex quadratic programs and linear-constrained robust nonlinear least squares are conducted to demonstrate the empirical efficiency of the proposed DPALM over several state-of-the-art methods and a specialized solver. Our code can be accessed at https://github.com/RPI-OPT/DampedALM.
