Suppose we want to find coefficients $b$ in $\underset{b}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n | y_{i}-b_{1}x_{i}-b_{0}\mid$.
If we rewrite this problem in terms of linear programming, we can use two different ways:
1. $$min \displaystyle\sum\limits_{i=1}^n |z_{i}|$$
subject to $z_{i}>=y_{i}-b_{1}x_{i}-b_{0}$; $z_{i}>=-y_{i}+b_{1}x_{i}+b_{0}$
or another way:
. $$min \displaystyle\sum\limits_{i=1}^n |z_{i}^{+}+z_{i}^{-}|$$
subject to $z_{i}^{+}-z_{i}^{-}=y_{i}-b_{1}x_{i}-b_{0}$
My question is which way is "better" in terms of computational performance for a standard linear solver for large number of b coefficients and observations.
