Linear programming with absolute values

I know that sometimes we can use absolute values into the objective functions or constraints. Is it always possible to use them, anywhere ?

Example of use of absolute values:

Minimize |a+b+c| + |a-c| s.t.
 |a| + b > 3
 | |a| - |b| | <= 5 
 | |b| - 3 | = 0

All constraints in a linear program are convex (if $x,y$ satisfy the constraints, then $tx+(1-t)y$ also does for all $0 \leq t \leq 1$). The constraint $|a|+b > 3$ is not convex, since $(4,0)$ and $(-4,0)$ are both solutions while $(0,0)$ is not. It is also not closed, which is another reason why you cannot use it in a linear program (change $>$ to $\geq$). The constrict $|a|+b \leq 3$, however, can be used, since it is equivalent to the pair of constraints $a+b \leq 3$ and $(-a)+b \leq 3$.

So absolute values can sometimes be expressed in the language of linear programming, but not always.

StackExchange Computer Science Q#44705, answer score: 10