Can we write
min($\sum_{i}$ max($a^T_i$x - $b_i$,0))
as a constrained optimization problem with a differentiable objective function assuming everything is real-valued. My understanding is that this is a 'known' problem in optimization, but I haven't found a good description of how this works.
Here is an equivalent linear program: $P_{LP}: \ \min \{ \sum_k m_k | m_k \ge a_k^T x -b_k, m_k \ge 0 \}$.
Let $P_O$ denote the original problem.
Suppose $(x,m)$ solves $P_{LP}$, then $m_k \ge 0 $ and $m_k \ge a_k^T x -b_k$ and hence $m_k \ge \max(a_k^T x -b_k,0)$. In fact, we must have $m_k = \max(a_k^T x -b_k,0)$ and so $x$ solves $P_O$.
If $x$ solves $P_{O}$, let $m_k = \max(a_k^T x -b_k,0)$, then $(x,m)$ solves $P_{LP}$.