algorithms – Linear programming reduce errors

Most normal linear programming problems look like this:
normal linear programming
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:
weird linear programming
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?