After decoupling a big optimization, the inner one is as follow:
$min_X \quad log_2 \left(det(A+BX\right))$
$s.t. norm(X)\leq \gamma,$
where $A\in \mathbb{C}^{n\times n}$ and $B\in \mathbb{C}^{n\times m}$ are known, while $X\in \mathbb{C}^{m\times n}$ includes optimization variables. As you can see the problem became minimization of a concave function. To avoid search on boundaries, I decided to linearize $det\left(A+BX\right)$ and ignore $log_2()$.
I appreciate any hint about linearization of $det\left(A+BX\right)$ in its general form without any assumption on $A$, $B$, or $BX$.