A Proximal Augmented Lagrangian Method for Linearly Constrained Nonconvex Composite Optimization Problems

This paper proposes and establishes the iteration complexity of an inexact proximal accelerated augmented Lagrangian (IPAAL) method for solving linearly constrained smooth nonconvex composite optimization problems. Each IPAAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a suitable Lagrange multiplier update. For any given (possibly infeasible) initial point and tolerance ρ >0 , it is shown that IPAAL generates an approximate stationary solution in 𝒪(ρ ^-3log (ρ ^-1)) ACG iterations, which can be improved to 𝒪(ρ ^-2.5log (ρ ^-1)) if it is further assumed that a certain Slater condition holds.