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The classical convergence theory of the augmented Lagrangian method has been developed under the assumption that the solutions satisfy a constraint qualification. The point of this note is to show that the constraint qualification can be limited to the constraints that are not enforced by the Lagrange multipliers. In particular, it follows that if the feasible set is non-empty and the inequality constraints are convex and separable, then the convergence of the algorithm is guaranteed without any additional assumptions. If the feasible set is empty and the projected gradients of the Lagrangians are forced to go to zero, then the iterates are shown to converge to the nearest well posed problem.

On Convergence of Inexact Augmented Lagrangians for Separable and Equality Convex QCQP Problems without Constraint Qualification