A sequential optimality condition related to the quasi-normality constraint qualification and its algorithmic consequences

Abstract

In the present paper, we prove that the augmented Lagrangian method converges to KKT points under the quasi-normality constraint qualification, which is associated with the external penalty theory. An interesting consequence is that the Lagrange multiplier estimates computed by the method remain bounded in the presence of the quasi-normality condition. In order to establish a more general convergence result, 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. PAKKT points are substantially better than points obtained by the classical approximate KKT (AKKT) condition, which has been used to establish theoretical convergence results for several methods. In particular, we present a simple problem with complementarity constraints such that all its feasible points are AKKT, while only the solutions and a pathological point are PAKKT. This shows the efficiency of the methods that reach PAKKT points, particularly the augmented Lagrangian algorithm, in such problems. We also provide the appropriate strict constraint qualification associated with the PAKKT sequential optimality condition, called PAKKT-regular, and we prove that it is strictly weaker than both quasi-normality and the cone continuity property. PAKKT-regular connects both branches of these independent constraint qualifications, generalizing all previous theoretical convergence results for the augmented Lagrangian method in the literature