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I am having trouble converting the following objective function into LP: $$ \min\left\lvert(\left\lvert x_1-a_1\right\rvert-\left\lvert x_2-a_2 \right\rvert )\right\rvert$$ where $x$ is the decision variable and $(a)$ is an integer constant.

I tried adding the following constraints, but it did not work out:

$x_i-a_i\le y_i$

$a_i-x_i\le y_i$

$y_1-y_2\le U$

$y_2-y_1\le U$

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  • $\begingroup$ Why it didn't work out? The only step missing is $\min U$, since we have $y_i \ge 0$. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Jan 29 '18 at 1:42
  • $\begingroup$ I think the inner absolute values need to be modeled with binary variables (that part is non-convex). $\endgroup$ – Erwin Kalvelagen Jan 29 '18 at 19:40
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I think this needs to be modeled as a MIP:

\begin{align} \min\> & z\\ & -z \le y_1 - y_2 \le z\\ & y_i \ge x_i - a_i\\ & y_i \ge -(x_i - a_i)\\ & y_i \le x_i - a_i + \delta_i M\\ & y_i \le -(x_i - a_i) + (1-\delta_i) M\\ & \delta_i \in \{0,1\} \end{align}

where $M$ is large enough constant.

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