# Optimizing with Absolute Value Objective Function

max : $w = |q^T y|$
subject to
$A y \leq b$
$y \geq 0$

Please describe how one could solve the non-linear programming prob. above by using linear programming methods.

I tried changing $y$ to $y' - y''$ in the constraints and $y' + y''$ for the objective function. However, my Excel solver says that "the cells do not converge". How should I solve this?

Thanks a bunch!

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the maximum value of $w$ may be infinite. do you have any information about $A,b,q$? – Ilya Oct 10 '11 at 13:42
no, but there's a hint saying: "Try breaking it into 2 linear programming problems. Then, could you think of combining them into just 1 problem?" – John Oct 10 '11 at 13:46

## 2 Answers

To follow the advise given to you, consider two problems: $$\begin{cases} w^+ &= q^Ty^+\to\max, \\ Qy^+&\leq0, \\ Ay^+&\leq b, \\ y^+&\geq 0. \end{cases}$$ and

$$\begin{cases} w^- &= q^Ty^-\to\max, \\ Qy^-&\geq0, \\ Ay^-&\leq b, \\ y^-&\geq 0. \end{cases}$$

Then $w = \max\{w^+,w^-\}$. Here matrix $Q = (q\quad0\quad\dots\quad0)^T$. With such decomposition you just consider two possible cases for the absolute value.

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Thanks for your answer! However, is there any way to solve this using just 1 linear programming problem? – John Oct 10 '11 at 17:05
@John: I'm afraid, I don't know such a way. – Ilya Oct 10 '11 at 17:29

Hint: let $y = y_1 + y_2$ where $q^Ty_1 \le 0$ and $q^Ty_2 \ge 0$.

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