Initially I modeled my objective function as
$$\arg \min \operatorname{Var}(f(x),g(x)) + \operatorname{Var}(c(x),d(x)) + \cdots$$
where $f$, $g$, $c$, $x$ are linear functions.
To be able to solve the problem with linear solvers I have change the problem as follows ($\operatorname{abs}$ stands for absolute value):
$$\arg \min\; \operatorname{abs}(f(x)-g(x)) + \operatorname{abs}(c(x)-d(x)) + \cdots$$
Is this form correct and has the same meaning as the first model?