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 pointsunder the quasinormality constraint qualification, which is associated with the external penalty theory. An interesting consequence is that the Lagrange multipliers estimates computed by the methodremain bounded in the presence of the quasinormality 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 optimalityconditions proposed by Hestenes, and later extensively treated by Bertsekas. PAKKT points aresubstantially 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 areAKKT, while only the solutions and a pathological point are PAKKT. This shows the efficiency of themethods that reach PAKKT points, particularly the augmented Lagrangian algorithm, in such problems. We also provided the appropriate strict constraint qualification associated with the PAKKTsequential optimality condition, called PAKKT-regular, and we prove that it is strictly weaker thanboth quasinormality and cone continuity property. PAKKT-regular connects both branches of theseindependent constraint qualifications, generalizing all previous theoretical convergence results for theaugmented Lagrangian method in the literature.