Converting nested absolute value into linear programming
This sounds like a homework problem, so just hints here.

We'd usually just transform $x_1$ and $x_2$ into two new variables,
$$
\begin{array}{ccccc}
x_1^{'}~{\equiv}~x_1-a_1 &amp; &amp; \text{and} &amp; &amp;
x_2^{'}~{\equiv}~x_2-a_2
\end{array}
,
$$such that the new problem is$$
\min{\left|
\begin{array}{c}
{\left|x_1^{'}\right|}-{\left|x_2^{'}\right|}
\end{array}
\right|}.
$$
Then, the absolute values imply symmetries such that we don't need to consider some parts of the domain.

We may ignore:


$x_1^{'}&lt;0$, because it's degenerate with $x_1^{'}&gt;0$ under $\left|x_1^{'}\right|$.
$x_2^{'}&lt;0$, because it's degenerate with $x_2^{'}&gt;0$ under $\left|x_2^{'}\right|$.
$\left|x_1^{'}\right|&lt;\left|x_2^{'}\right|$, because it's degenerate with $\left|x_1^{'}\right|&gt;\left|x_2^{'}\right|$ under $\left|\left|x_1^{'}\right|-\left|x_2^{'}\right|\right|$.


Typically you can specify these sorts of domains to ignore in terms of constraints.

Then, since the degenerate cases have been removed, the objective function's just$$
\min{
\left(
{x_1^{'}}-{x_2^{'}}
\right)
}.
$$