In the present paper, we prove that the augmented Lagrangian method converges to KKT points under the quasinormality constraint qualification, which is associated with the external penalty theory. For this purpose, a new sequential optimality condition for smooth constrained optimization, called PAKKT, is defined. The new condition takes into account the sign of the dual sequence, constituting an adequate sequential counterpart to the (enhanced) Fritz-John necessary optimality conditions proposed by Hestenes, and later extensively treated by Bertsekas. We also provided the appropriate strict constraint qualification associated with the PAKKT sequential optimality condition and we prove that it is strictly weaker than both quasinormality and cone continuity property. This generalizes all previous theoretical convergence results for the augmented Lagrangian method in the literature.
Institute of Mathematics, University of Campinas, Brazil; CONICET, Department of Mathematics, FCE, University of La Plata, Argentina; Department of Applied Mathematics, Federal University of Espírito Santo, Brazil, September 2017.