$L_1$ norm regularization via gradient descent solvable in order to find a minimal solution?

Question

Do I find a (of many) solution with the gradient descent, if

$$a^Tx=b,$$ $a=(-1,1)$ and $b = 2$

For example: $Ax=b \rightarrow (-1 \; 1)\cdot (x_1 \; x_2)^T = 2$

I set $x_1$ to -1 and $x_2$ to 1:

$-1 + x_2 = 2 \rightarrow x_2 = 3$

$-x_1 + 1 = 2 \rightarrow x_1 = 1$

Then I get a possible solution $ x = (3 \; 1)^T $, but I think that it is not a minimum one.