Given a system of linear equations over $\mathbb{R}^n$ $$Ax=b$$, where $A \in \mathbb{R}^{m \times n}, x \in \mathbb{R}^n, b \in \mathbb{R}^m, m >n$, I want to minimize the following objective, which is intuitively least-squares but with the $L_1$ norm: $$ \min_{x} ||Ax-b||_{1}$$
This can clearly be solved via linear programming as laid out here, but I am interested in a closed form solution to the problem, as can be found found in the least-squares case. Is such a closed form solution known?
