Consider two systems A and B:
A$: f_i(x) \lt 0, i=1, \dots, m, Ax = b$
B$: \min_{\{x,s\}} s$ subject to $f_i(x) - s \le 0, i = 1, \dots, m; Ax = b$
Then B has optimal value $p^* \lt 0$ iff there exists a solution to A
I can prove $\Rightarrow$ but I'm having trouble showing $\Leftarrow$.
My original attempt was to assume $x$ is the solution of A, and therefore: IF $x$ is used in B then the smallest $s$ that satisfies the equation must be in $[f_i(x), 0)$ since $f_i(x) \lt 0$. But this is assuming that for $B$ the same value of $x$ is used. Which doesn't seem right because $x$ might not work for B.
Any ideas?