Reviews: An Inexact Augmented Lagrangian Framework for Nonconvex Optimization with Nonlinear Constraints

Applying ALM to the Burer-Monterio problem and to nonlinear programs in general is natural and well summarized in the monograph Ref [8]. Allowing first-order and second-order approximate solvers for the primal subproblems is also classic, and can be found in, e.g., Ch 8 & 9 of Ref [8]. I think the main novelties here lie at the nonsmooth, convex term g(x) and the convergence rate results. Sec 5 of the paper has provided a comprehensive review of pertinent results under different assumptions. I have several concerns that I hope the authors can address: * The BM example does not quite justify the inclusion of the possibly nonsmooth term g in (1). The authors may want to balance out and briefly discuss other examples as appearing in the experiments.