Convert Sums of absolute values in Linear program Ihave the following problem and it needs to be converted into LP form.
\begin{align}
 \max\&gt; &amp; z = c^T x\\
      &amp; Ax \leq B \\
      &amp; \sum_{i=1}^{n}|x_i|=1
\end{align}
I know that $c^Tx$ and $Ax \leq B$ are matrix dot products.
My question is, How can I convert $\sum_{i=1}^{n}|x_i|=1$ into LP form.
I have read online that if you only have a finite amount of $x$'s, you can simply do one equation for the positive and another one for the negative. For instance, if we have $|x_1| = 1$, the equation will be transformed to $x_1 = 1$ and $-x_1 = 1$. How can we do it for the summation?
I think we need to take the combinations of all the possible $x$'s as maybe $x_1$ is positive and all the rest are negative.
Thanks!
Use auxiliary variables. Let $u_i \geq 0, v_i \geq 0$. Then we can write $x_i = u_i - v_i$ and $\vert x_i \vert = u_i + v_i$.
Use auxiliary variables. Let $u_i \geq 0, v_i \geq 0$. Then we can write $x_i = u_i - v_i$ and $\vert x_i \vert = u_i + v_i$.