  This work presents an adaptive superfast proximal augmented Lagrangian
(AS-PAL) method for solving linearly-constrained smooth nonconvex composite
optimization problems. Each iteration of AS-PAL inexactly solves a possibly
nonconvex proximal augmented Lagrangian (AL) subproblem obtained by an
aggressive/adaptive choice of prox stepsize with the aim of substantially
improving its computational performance followed by a full Lagrangian
multiplier update. A major advantage of AS-PAL compared to other AL methods is
that it requires no knowledge of parameters (e.g., size of constraint matrix,
objective function curvatures, etc) associated with the optimization problem,
due to its adaptive nature not only in choosing the prox stepsize but also in
using a crucial adaptive accelerated composite gradient variant to solve the
proximal AL subproblems. The speed and efficiency of AS-PAL is demonstrated
through extensive computational experiments showing that it can solve many
instances more than ten times faster than other state-of-the-art penalty and AL
methods, particularly when high accuracy is required.
