Before revision:  To solve our estimation problems in a convex framework, we apply alternating direction method of multiplier (ADMM) which is shown to be suitable for a large-scale implementation. In combination with applying model selection criteria, the penalty parameter in the regularized estimation, controlling the density of nonzero entries in the path matrix, can be chosen to provide a reasonable trade-off between the model fitting and the complexity of causality structure.
After revision:  To solve our estimation problems in a convex framework, we apply alternating direction method of multiplier (ADMM) which is shown to be suitable for a large-scale setting.