For dual methods of multilevel optimization local applicability conditions which secure both the existence of unique local solutions of the subproblems and the validity of the strong local duality are given. Using an augmented Lagrangian function the reduction of the conditions to the well-known sufficient optimality conditions and the strict complementarity assumption is possible. Connections of the local duality with a local perturbation theory are shown, and properties of the dual problem and the perturbation function considered.