I generalize the problem to be:
Let $P = \{\mathbf{A}\mathbf{x} \leq \mathbf{b}: x \in \mathbb{R}^d\}$ such that we want to minimize $\sum_{i=1}^n f(x_i)$ where
$$ f(x_i) := \begin{cases} 10x_i, ~~~x_i < 0 \\ 100x_i, ~~~x_i \geq 0. \end{cases}$$
Further, suppose $\mathbf{A}$ unknown. Instead, for known vectors $\{\alpha_i\}_{i \in \mathbb{N}}^n$ and $\{c_i\}_{i \in \mathbb{N}}^n$ we have that
$$||a_i - \alpha_i||_{\infty} \leq c_i. \tag{1}$$
A remark mentions that (1) implies $a_{ij} \in [\alpha_{ij} - c_i, \alpha_{ij} + c_i]$, and then to just define some auxiliary variables to get a linear program. This doesn't quite seem obvious to me so I tried to construct it:
$$ \begin{aligned} & \underset{}{\min} && \sum_{i=1}^n z_i \\ & \text{subject to} & & z_i \geq 10x_i \\ &&& z_i \geq 100x_i \\ &&& y_i \geq a_{ij} - \alpha_{ij} \\ &&& y_i \geq \alpha_{ij} - a_{ij} \\ &&& y_i \leq c_i \\ &&& \mathbf{y}'\mathbf{z} \leq \mathbf{b} \\ &&& \hspace{-1cm} -\infty < a_{ij} < \infty \end{aligned} $$
For all $i = 1, \dots, n$ and $j \in \mathbb{N}$. Would this be a correct formulation? Specifically, I'm unsure of how I modeled $y_i$, and if this is proper:
$$y_i = ||a_i - \alpha_i||_{\infty} = \max\{|a_{ij} - \alpha_{ij}| : \forall j \in \mathbb{N}\}.$$