A note on the KL property of the augmented Lagrangian for conic programming

The relationship between the Kurdyka-Łojasiewicz (KL) property of the augmented Lagrangian for conic programming and various calmness conditions of its perturbed Karush-Kuhn-Tucker (KKT) system solution mapping is studied in this paper. We establish equivalences among calmness conditions —specifically calmness, semi-isolated calmness, and isolated calmness—between the perturbed KKT system solution mapping and the inverse of the subdifferential of the augmented Lagrangian. Under certain conditions, we demonstrate that calmness of the perturbed KKT system solution mapping implies the KL property of its augmented Lagrangian with an exponent of 1/2 at KKT points. Moreover, both semi-isolated calmness and isolated calmness of the KKT system solution mapping directly imply that the augmented Lagrangian satisfies the KL property with an exponent of 1/2 at KKT points. These results provide several approaches to determine the KL property of the augmented Lagrangian at KKT points, and their application is illustrated through examples.