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Here is the least absolute deviation problem under concerned: $ \underset{\textbf{w}}{\arg\min} L(w)=\sum_{i=1}^{n}|y_{i}-\textbf{w}^T\textbf{x}|$. I know it can be rearranged as LP problem in following way:

$\min \sum_{i=1}^{n}u_{i}$

$u_i \geq \textbf{x}^T\textbf{w}- y_{i} \; i = 1,\ldots,n$

$u_i \geq -\left(\textbf{x}^T\textbf{w}-y_{i}\right) \; i = 1,\ldots,n$

I have no idea to solve it with simplex method. I don't want to use package but solve it with written computation. Could you please help me with this problem? Thanks in advance!

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    Do want the minimum or maximum (as you've written) of the objective function? The maximization problem is non-convex and can't be formulated as an LP. – 2017-02-06
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    @BrianBorchers Thanks for your reply. Sorry it was a mistake. I have edited it. – 2017-02-06
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    Your LP is incorrect: you need two constraints per $u_i$. Do you really want to solve a problem with $2n$ constraints by hand? – 2017-02-07
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    @LinAlg Thanks for your reply. But what do you mean by $2n$ constraint? thanks a lot – 2017-02-07
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    If $n=10$ you end up with $20$ constraints. – 2017-02-07
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    @LinAlg indeed I am a newbie to LP, I think I may make the problem more complex. Do you have better idea? Thanks! – 2017-02-07
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    If $w$ is a vector, this is not a problem to solve on paper. – 2017-02-07
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    @LinAlg $x$ is a vector too. – 2017-02-07
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    @LinAlg And $w$ is a vector – 2017-02-07

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