Infeasible linear programming reduce errors to find solution Most normal linear programming problems look like this:

We choose some point in the double shaded area that solves our optimizer and we're good to go.
However, I've come across a problem where it's fairly common to get a lp problem like this (infeasible):

I'd like to employ some metric such that the solution given reduces the errors between the closest possible solutions in any equation it does not fulfill such that it would be on the line right between the 2 solutions (something like reduce OLS or LAD).
Is there any formal algorithm for this or common way this is done?
One approach: replace each inequality $a_1 x_1 + \dots + a_n x_n \le b$ by $a_1 x_1 + \dots + a_n x_n \le b + t$, replace each inequality $a_1 x_1 + \dots + a_n x_n \ge b$ by $a_1 x_1 + \dots + a_n x_n \ge b - t$ (using the same $t$ for all of them), and then minimize $t$.