How can I solve efficiently the problem:
$$\min_x \| A x \|_1 \quad \text{subject to} \; {b}^{T} x = c$$
for $A \in \mathbb{R}^{m \times n}, b \in \mathbb{R}^{n}, c \in \mathbb{R}$. The matrix $A$ has independent columns.
Is there a closed form solution? Or a trick to quickly calculate $x$?
I am aware it can be solved iteratively by projected gradient descent and by other alternating methods. Yet they converge pretty slowsly.