Given the Chebyshev (or minimax) approximation problem:
$\min_\limits{x} ||Ax-b||_\infty, \\ \text{subject to } x \succeq 0$
I want to write it as a linear program in the following manner:
$\min_\limits{y, t} t, \\ \text{subject to } a^T_{i}y -t \leq b_i \forall i=1,...k, \\ -a^T_{i}y -t \leq -b_i \forall i=1,...k, \\ y \succeq 0$
My initial approach is to take $y=x$ and $t = ||Ax-b||_\infty$ so that I can show that every feasible solution $x$ of the Chebyshev optimization problem has a feasible solution $(y, t)$ in the linear program, and vice versa. How do I proceed from here, or correct my approach to solving this?