I am trying to check if the following function is convex or not:
$$f(V) = \|A - BC(DBC)^{-1}\|^2_F$$
$$B=\exp\left( V F V^T \right)$$ $$F=\operatorname{diag}\left(f^1, \ldots , f^n\right)$$
where $f^i$ is a scalar for $i = 1$ to $n$ and $A \in\mathbb R^{n\times m}$, $D \in\mathbb R^{m\times n}$, $B \in\mathbb R^{n\times n}$, $C \in\mathbb R^{n\times m}$ and $V \in\mathbb R^{n\times n}$. I know that the norm is a convex function, but I am not able to check if the matrix inside the norm is convex/concave and increasing/decreasing. Any hints or directions will be much appreciated. Also, in case it is non-convex, is it possible to impose conditions on $V$ such that the function is convex.