Iteration-complexity of a proximal augmented Lagrangian method for solving nonconvex composite optimization problems with nonlinear convex constraints

This paper proposes and analyzes a proximal augmented Lagrangian (NL-IAPIAL) method for solving smooth nonconvex composite optimization problems with nonlinear cal K-convex constraints, i.e., the constraints are convex with respect to the order given by a closed convex cone cal K. Each NL-IAPIAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a Lagrange multiplier update. Under some mild assumptions, it is shown that NL-IAPIAL generates an approximate stationary solution of the constrained problem in cal O(log(1/ρ)/ρ3) inner iterations, where ρ>0 is a given tolerance. Numerical experiments are also given to illustrate the computational efficiency of the proposed method.