I need a linear program to minimize the sum of several absolute values, but the inclusion of an absolute value means the linear solver won't work. I know there are ways around using an absolute value, but none of the fixes I've seen apply when you're trying to minimize a sum of several absolute values.
Specifically, I have 3 sets of constants ($a_1, a2,\ldots$; $\ b_1, b_2,\ldots$; & $\ c_1, c_2,\ldots$) and 2 variables ($x$ & $y$). The gist of my program is below.
$$\min (|a_1 x + b_1 y - c_1| + |a_2 x + b_2 y - c_2| + |a_3 x + b_3 y - c_3|)$$
$$x + y = 1$$