I have the following optimization problem $$ \min_{q\in Q_0} \sum_{x=1}^X |m_x(q_{1})| $$ where
$q\equiv (q_1,q_2)$ is a vector that should satisfy a bunch of non-linear constraints collected in $Q_0$.
$m_x(q_1)$ is linear in $q_1$ $\forall x=1,...,X$.
Specifically: some components of $q_2$ are required to be binary, other are allowed to vary in $[l_b, l_u]$; all the elements of $q_1$ are allowed to vary in $[l_b, l_u]$; apart from such support restrictions, $Q_0$ contains inequalities of the type e.g., $$ \overbrace{q_{1,j}}^{\in [l_b,l_u]}-\overbrace{q_{1,k}}^{\in [l_b,l_u]}\geq \overbrace{q_{2,h}}^{\text{binary}} $$ where $q_{1,j}$ is the $j$th element of $q_1$.
In other words, what I want to highlight (sorry for the non-formal, maybe wrong terminology) is that all the inequalities in $Q_0$ would have been linear if all the elements in $q_2$ were allowed to freely vary in $[l_b, l_u]$.
Question: I have been advised that the problem above can be rewritten as having a linear objective function, because it is equivalent to $$ \begin{aligned} &\min_{q, \{\mu^{+}(x), \mu^{-}(x)\}_{\forall x =1,...,X}} \sum_{x=1}^X (\mu^{+}(x)+\mu^{-}(x))\\ & q\in Q_0\\ & \mu^{+}(x)\geq 0 \text{ }\forall x=1,...,X\\ & \mu^{+}(x)\geq 0\text{ }\forall x=1,...,X\\ & m_x(q)=\mu^{+}(x)-\mu^{-}(x)\text{ }\forall x=1,...,X\\ \end{aligned} $$ Is this correct? If yes, could you skectch a proof? If not, why?
My impression is that the reformulation of the problem works when $Q_0$ contains constraints that are linear in $q$ (e.g., p.294 of Boyd and Vandenberghe, 2004). When $Q_0$ has non-linear constraints (as in my case) I have doubts.