$$\begin{array}{ll} \underset{x \in \mathbb{R}^4}{\text{minimize}} & x_1 + 6 x_2 - \min\{10x_3, 5x_4\} + \left| \displaystyle\sum_{i=1}^{4}x_i \right| \\ \text{subject to} & \displaystyle\sum_{i=1}^{4}x_i \ge \left| \displaystyle\sum_{i=1}^{3} i x_i \right| \\ & x_1, x_2, x_3, x_4 \ge 0 \end{array}$$
My solution:
First let $z=|\sum_{i=1}^{4}x_i|$, and let $t=\min\{10x_3,5x_4\}$. Now we get the following problem:
$\min_{x\in\mathbb{R}^4,z}x_1+6x_2-t+z$.
S.t.
$\sum_{i=1}^{4}x_i\ge |\sum_{i=1}^{3}i*x_i|$.
$z=|\sum_{i=1}^{4}x_i|$.
$t\le 10x_3$, $t\le 5x_4$.
$x_i\ge 0, \forall i=1,2,3,4$.
Now, We notice that for every optimal solution $z$, there must be equality in the constraint $z=|\sum_{i=1}^{4}x_i|$, (we can assume that there exists an optimal solution $z$ such that $z>|\sum_{i=1}^{4}x_i|$, but that means that there exists $\epsilon>0$ such that $z-\epsilon$ lowers our function value, and we get contradiction that $z$ is an optimal solution). So we can write: $z\ge|\sum_{i=1}^{4}x_i|$ and still keep the same problem.
And we have:
$\min_{x\in\mathbb{R}^4,z}x_1+6x_2-t+z$.
S.t.
$\sum_{i=1}^{4}x_i\ge \sum_{i=1}^{3}i*x_i$.
$\sum_{i=1}^{4}x_i\le -\sum_{i=1}^{3}i*x_i$.
$z\ge \sum_{i=1}^{4}x_i$.
$z\le -\sum_{i=1}^{4}x_i$.
$t\le 10x_3$, $t\le 5x_4$.
$x_i\ge 0, \forall i=1,2,3,4$.
And that's my final answer.
Little question: Do I need to add $z$ to $\min_{x\in\mathbb{R}^4,z}$? why?
Any feedback is really appreciated, thanks in advance!
EDIT:
I've noticed that I have no logic behind changing $t = \min\{10x_3,5x_4\}$ with $t\le 10x_3$ and $t\le 5x_4$, Since this is a minimum problem, I'm not sure that the optimal solutions are the same! Any help?
You have two errors (essentially the same error twice): in the second constraint for linearizing each absolute value, the $\le$ should instead be $\ge$.
Also, if the problem really imposes that each $x_i \ge 0$, you can simply remove the absolute values. For example, replace $|\sum_i x_i|$ with just $\sum_i x_i$ because these two expressions are equal.