I have a list of equations that define halfspaces (3 equation list example below)
$x_1 + x_2 + x_3 + x_5 < x_4 + x_6 + x_7 \\ x_4 + x_6 + x_7 < x_8 + x_9 + x_{21} + x_{43} + x_1 \\ x_4 + x_6 + x_7 < x_1 + x_2 + x_3 + x_5$
As shown, sometimes this system will not be solvable (I've shown a simple case; in n-dim, it's obvious how this may happen in a more complex way). In this case, I want to find the values of all $x_n$ that has the lowest euclidean distance from being viable.
I'm not entirely sure if I want euclidean distance or squared euclidean distance., but my first thoughts jump to OLS, LAD, and linear programming. However, I've always seen OLS/LAD done with a bunch of real number valued points, not to solve a linear programming system.
I also got informed that this may be reformulateable as a classic nonlinear (specifically quadratic) programming problem, although I'm unfamiliar with the space as a whole.
Any ideas on how to best find the values of this system.?