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This paper proposes and establishes the iteration-complexity of an inexactproximal accelerated augmented Lagrangian (IPAAL) method for solving linearlyconstrained smooth nonconvex composite optimization problems. Each IPAALiteration consists of inexactly solving a proximal augmented Lagrangiansubproblem by an accelerated composite gradient (ACG) method followed by asuitable Lagrange multiplier update. It is shown that IPAAL generates anapproximate stationary solution in at most ${\cal O}(\log(1/\rho)/\rho^{3})$ACG iterations, where $\rho>0$ is the given tolerance. It is also shown thatthe previous complexity bound can be sharpened to ${\calO}(\log(1/\rho)/\rho^{2.5})$ under additional mildly stronger assumptions. Theabove bounds are derived assuming that the initial point is neither feasiblenor the domain of the composite term of the objective function is bounded. Somepreliminary numerical results are presented to illustrate the performance ofthe IPAAL method.

Iteration-complexity of an inexact proximal accelerated augmented  Lagrangian method for solving linearly constrained smooth nonconvex composite  optimization problems