  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 ${\cal O}(\log(1/\rho)/\rho^{3})$ inner iterations, where $\rho>0$
is a given tolerance. Numerical experiments are also given to illustrate the
computational efficiency of the proposed method.
