Unified theory of augmented Lagrangian methods for constrained global optimization

We classify in this paper different augmented Lagrangian functions into three unified classes. Based on two unified formulations, we construct, respectively, two convergent augmented Lagrangian methods that do not require the global solvability of the Lagrangian relaxation and whose global convergence properties do not require the boundedness of the multiplier sequence and any constraint qualification. In particular, when the sequence of iteration points does not converge, we give a sufficient and necessary condition for the convergence of the objective value of the iteration points. We further derive two multiplier algorithms which require the same convergence condition and possess the same properties as the proposed convergent augmented Lagrangian methods. The existence of a global saddle point is crucial to guarantee the success of a dual search. We generalize in the second half of this paper the existence theorems for a global saddle point in the literature under the framework of the unified classes of augmented Lagrangian functions.