linear programming : Absolute value in constraint in mathematical model

Question

I have a model have an constraint with evaluation of absolute value , a example can be:

function objective : $\max \sum(x_i)$ statement: $x_i\geq |(y_i-t_i)|$ for all $i$

but value absolute not is permitted in solver complex , the model is non-convex

Answer 1

Write the inequality with the absolute value in two seperate inequalities. First define

$y_i-t_i=a_i$

$t_i-y_i=b_i$

Then $x_i$ have to be larger or equal to $a_i$ and $b_i$:

$x_i\geq a_i$

$x_i \geq b_i$

$\texttt{numerical example:}$

Case 1: Suppose $y_i=3$ and $t_i=2$. $a_i$ becomes 1 and $b_i$ becomes -1. Therefore $x_i$ has to be greater or equal than 1.

Case 2: Suppose $y_i=2$ and $t_i=3$. $a_i$ becomes -1 and $b_i$ becomes 1. Therefore $x_i$ has to be greater or equal than 1.

In both cases $x_i$ has to be greater or equal than 1.