Comparison of Two Decomposition Approaches with the Augmented Lagrangian Function

The decomposition methods are used to solve large-scale optimization problems by decomposition them into sub-problems.The main drawback of the augmented Lagrangian relaxation method is that the quadratic term introduced by the augmented Lagrangian is not separable.To cope with the non-separability of the augmented Lagrangian function,we can apply auxiliary problem principle(APP) method or block coordinate descent(BCD) method to the augmented Lagrangian relaxation method.Compared with the literature in solving the optimization problem with constraints x-x=0,we compare these two decomposition methods solving optimization problem with more general constraints——linear constrains z=Ax.Two numerical examples are to show comparison of theoretical validation—In dealing with non-separable augmented Lagrangian function,we can often expect faster performance of the BCD method compared to the APP method under certain conditions.