  The augmented Lagrangian method (ALM) is classic for canonical convex
programming problems with linear constraints, and it finds many applications in
various scientific computing areas. A major advantage of the ALM is that the
step for updating the dual variable can be further relaxed with a step size in
$(0,2)$, and this advantage can easily lead to numerical acceleration for the
ALM. When a separable convex programming problem is discussed and a
corresponding splitting version of the classic ALM is considered, convergence
may not be guaranteed and thus it is seemingly impossible that a step size in
$(0,2)$ can be carried on to the relaxation step for updating the dual
variable. We show that for a parallel splitting version of the ALM, a step size
in $(0,2)$ can be maintained for further relaxing both the primal and dual
variables if the relaxation step is simply corrected by a rank-two matrix.
Hence, a rank-two relaxed parallel splitting version of the ALM with a step
size in $(0,2)$ is proposed for separable convex programming problems. We
validate that the new algorithm can numerically outperform existing algorithms
of the same kind significantly by testing some applications.
