How to minimize $|x_1| + |x_2|$ via linear programming?

Question

Given $a_{ij}, b_j \in \Bbb R$, consider the following optimization problem:

$$\begin{array}{ll} \underset{x_1, x_2 \in \Bbb R}{\text{minimize}} & |x_1| + |x_2|\\ \text{subject to} & a_{11}x_1 + a_{12}x_2 = b_1\\ & a_{21}x_1 + a_{22}x_2 = b_2\end{array}$$

Can I solve this problem with linear programming methods? If so, how?

This system does not have a guaranteed solution, is that ok? — user130512, Feb 26 '14 at 16:07
yes, for example if $a_{11}=a_{21}$ and $a_{12}=a_{22}$ but $b_1 \neq b_2$. — Stefano, Feb 26 '14 at 17:18
https://math.stackexchange.com/questions/432003/converting-absolute-value-program-into-linear-program — , Dec 17 '17 at 13:29

score 3 · Accepted Answer · answered Feb 26 '14 at 16:12

3

You can! Set $x_1=x_1^+ - x_1^-$ where $x_1^+ \geq 0$ and $x_1^- \geq 0$ then $|x_1|=x_1^+ + x_1^-$. Same for $x_2$.

This works since it is a minimization problem.

answered Feb 26 '14 at 16:12

Stefano

386

1

To be more clear: every real number can be written as a difference. If $x_1$ is negative the problem is going to set $x_1^-=x_1$ and $x_1^+=0$ since it wants to minimize their sum. – Stefano Feb 26 '14 at 16:16

score 1 · Answer 2 · answered Feb 26 '14 at 16:10

You can divide the problem up in four parts. One part is minimizing $x_1 + x_2$ subject to your constraints and $x_1, x_2 \geq 0$.

Another part would be minimizing $-x_1 + x_2$ subject to your constraints and $x_1 \leq 0$, $x_2 \geq 0$, and so on. One part for each quadrant.

Each part can be solved by linear programming methods, and the lowest minimum of all is the minimum of the original problem.

How to minimize $|x_1| + |x_2|$ via linear programming?

2 Answers2