$ -t \leq a_i^Tx - b_i \leq t$ being true for all i is equivalent to $\max\limits_i |a_i^Tx - b_i| \leq t$

Minimizing t then drives $| Ax - b \|_\infty$ as small as possible. It's really that simple.

$ -t \leq a_i^Tx - b_i \leq t$ being true for all i is equivalent to $\max\limits_i |a_i^Tx - b_i| \leq t$

Minimizing t then drives $\| Ax - b \|_\infty$ as small as possible. It's really that simple.