What is the best method for solving the following convex regularization problem?
$$ \text{minimize} ~~ c^T x + \| A x \|_{1} \quad \text{subject to} \quad -1 \leq x_i \leq 1 $$
where $c$ is a vector and $A$ is a matrix.
The original problem is a non-convex problem, but when I apply my algorithm the subproblems of my algorithm come out convex in the form of above. So I am trying to solve the above subproblem by convex optimization techniques.
It can be made into a linear programming problem using some auxiliary variables:
minimize $c^T x + \sum_j t_j$ subject to $-1 \le x_i \le 1$ for each $i$, $(Ax)_j \le t_j$ and $-(Ax)_j \le t_j$ for each $j$.