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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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    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?"2011-10-10

2 Answers 2

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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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    @John: I'm afraid, I don't know such a way.2011-10-10
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Hint: let $y = y_1 + y_2$ where $q^Ty_1 \le 0$ and $q^Ty_2 \ge 0$.