TITLE: Minimizing the sum of absolute values with a linear solver
QUESTION [6 upvotes]: 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.
$$\min (|a_1 x + b_1 y - c_1| + |a_2 x + b_2 y - c_2| + |a_3 x + b_3 y - c_3|)$$
such that  
$$x + y = 1$$

REPLY [10 votes]: This is really a comment on littlO's answer. Comment box is too small for this.
$$
\min (|a1 x + b1 y - c1| + |a2 x + b2 y - c2| + |a3 x + b3 y - c3|), x+y = 1$$
is equivalent to
$$
\min t_1 + t_2 + t_3  
\text{ such that} \\
a_1 x + b_1 y -c_1 \le t_1 \\
a_1 x + b_1 y -c_1 \ge -t_1 \\
a_2 x + b_2 y -c_2 \le t_2\\
a_2 x + b_2 y -c_2 \ge -t_2 \\
a_3 x + b_3 y -c_3 \le t_3 \\
a_3 x + b_3 y -c_3 \ge -t_3 \\
x+y = 1
$$
This just expands on littleO's answer. Please give littleO the credit.
Edited: (typo revised) from: $$a_3 x + b_3 y -c_3 \le t_1 \\$$ to: $$a_3 x + b_3 y -c_3 \le t_3 \\$$