Original program: $$\min (c^T x) \text{ subject to } Ax \le b \text{ for all $A \in \Bbb A$}$$ where $\Bbb A = \{A \in R^{m \times n} : |A_{ij} - \bar A_{ij} | \le V_{ij}, i = 1, \dots, m, j = 1, \dots, n\}$
Then this is equivalent to
$$\min (c^Tx) \text{ subject to } \bar Ax + V|x| \le b$$
where $|x| = (|x_1|, \dots, |x_n|)$
Can someone explain how the second program how the second program was derived?
What's the thought process behind $V|x|$?
It is easier if you look at just one constraint: $$a^Tx \leq b \quad \forall a \in A$$ $$\max_{a \in A} a^Tx \leq b$$ $$\max_{a : a = \bar{a} \pm v} a^Tx \leq b$$ Now distinguish between $x_i \leq 0$ vs $x_i > 0$ to decide between $\bar{a}+v$ and $\bar{a}-v$.