I would like to solve a linear system $Ax = b$ under the $L_1$ norm constraint $\min(|Ax-b|)$. All that I can find about $L_1$ minimization is a way to minimize $|x|_1$ subject to $Ax=b$.
I wanted to use linear programming in matlab to solve this problem. This lead me to solving a new system with linprog in matlab:
$$My = k$$
So I did some transformations, knowing that Linprog can solve the following problem:
$$\min f(x) \ \ \textrm{s.t.} \ \ Ax \leq b$$
I want to minimize this problem:
$$\min ||My - k||_1$$
for $y$
To convert this to the linprog form, I introduced a new vector I want to minimize, $z$:
$$\min \sum z$$
s.t.
$$My - k \leq z$$ $$-My + k \leq z$$
However, the constraints include both the variable $y$ and the constants $k$. We can formulate this in the required form by creating a bigger system:
$Ax \leq b$, where the matrix $A$ is:
$$\begin{pmatrix} M & -I \\ -M & I \end{pmatrix} $$
the vector $x$ is equal to $y$ and $z$ stacked: $$\begin{pmatrix} y \\ z \end{pmatrix} $$
and the constraints are $k$ and $-k$ stacked: $$\begin{pmatrix} k \\ -k \end{pmatrix} $$
The problem is that this doesn't work. (Matlab can't find a good solution). I would like to know if there is another way to solve this problem? If you have heard about a matlab code that already does it?
Thanks a lot