I have the following problem:
$$\min_{x\in X}\|Mx-c\|_{\infty}$$
I am considering a particular case, in which: $$M=\left[\begin{array}{cc} a & 1-a\\ b & 1-b \end{array}\right], c=\left[\begin{array}{c} A\\ B \end{array}\right] $$ where $0\leq a<b\leq1$ (so $\det(M)<0$ and thus $M$ is invertible), and $ 0\leq A<B\leq1$.
Also, $X$ is all the vectors whose elements are both in $[0,1]$. I have shown that the optimal unconstrained solution to $Mx=c$ lies outside $X$, so I know that this minimum is strictly positive.
My goal is to find a general upper bound for the above problem; i.e, an $\alpha$ (expressed in terms of the values $a, b, A, B$, which I don't know in advance) such that
$$\min_{x\in X}\|Ax-b\|_{\infty} \leq \alpha$$
I know that this problem can be formulated as an LP (see here), but I'm not sure if that buys me something - are there generic closed-form upper bounds for things that can be solved by an LP? Are there other approaches I can use? I am willing to assume more about the problem, in particular that the difference $B-A$ is somehow tied to the difference $b-a$.
Thanks!