I would like to find the optimal set $ \{ x_i \} $ given $ L $ and $ \{ a_i \} $ that minimizes the problem below. My first thought was to use linear programming. Is there a transformation that makes it possible, or do I need a more general optimization technique?
$$ \min_{x_i} \left[ 2\sum_i | x_i | + \max_i | x_i + a_i | \right]$$ $$ \mathrm{s.t.} \sum_i (x_i + a_i) \le L$$
You haven't told us what set $i$ ranges over, so I'll just assume $i=1, 2, \ldots, n$.
A standard trick in LP formulation of problems with absolute values is to introduce auxiliary variables and constraints with the basic idea that
$\min | x | $
is equivalent to
$\min t $
$t \geq x $
$t \geq -x $
Applying that idea to your problem, introduce auxiliary variables $t_{i}$, $i=1, 2, \ldots n$, and $s$. Then formulate the problem as:
$\min 2 \sum_{i=1}^{n} t_{i} + s $
subject to
$t_{i} \geq x_{i}$, $i=1, 2, \ldots n$.
$t_{i} \geq -x_{i}$, $i=1, 2, \ldots n$.
$s \geq x_{i}+a_{i} $, $i=1, 2, \ldots, n$.
$s \geq -(x_{i}+a_{i}) $, $i=1, 2, \ldots, n$.
$\sum_{i=1}^{n} (x_{i}+a_{i}) \leq L$
