How to find the optimal ${\bf Q}$ that maximizes $\left|{\bf I}+{\bf AQA^HB^{-1}}\right|$ I have the following question, which has stumped me for a long time.
${\max}_{\bf Q \succeq 0}~~~~{\rm det}\left({\bf I}+{\bf AQA^HB^{-1}}\right)$ 
s.t.$~~~{\rm Tr}\left({\bf Q}\right) \leq 1$.
where $\mathbf{B}=\operatorname{diag}\left[\mu_{1}, \cdots, \mu_{N}\right]$ is a diagonal matrix, and ${\bf A}$ is an arbitrary matrix.
Is there any possibility to find the optimal solution ${\bf Q}$ that has an explicit relationship with the diagonal elements of ${\bf B}$? (because $\left\{\mu_i\right\}$ are also variables.)
 A: At first glance I'd say that it looks difficult to find a closed form solution that depends directly on $B$. Using simple algebra, one can reshape the objective as:
${\rm det}\left({\bf I}+{\bf AQA^HB^{-1}}\right) = {\rm det}\left({\bf B}+{\bf AQA^H}\right) {\rm det}\left(\bf B^{-1}\right) = {\rm det}\left(\bf B^{-1/2}\right){\rm det}\left({\bf B}+{\bf AQA^H}\right) {\rm det}\left(\bf B^{-1/2}\right) = {\rm det}\left({\bf I}+{\bf B^{-1/2}AQA^HB^{-1/2}}\right)$ 
From here, write $C = B^{-1/2}A$, and write its SVD as $C = U_C \Sigma_C V_C^T$ and the SVD of $Q$ as $Q = U_Q \Sigma_Q U^T_Q$ (as Q is symmetric), then we have that the objective is:
${\rm det}\left( I + U_C \Sigma_C V_C^T U_Q \Sigma_Q U_Q^T V_C \Sigma_C U_C^T\right)$
We can take the orthonormal matrices $U_C$ out of the objective (as $I = U_C U_C^T$) and (and here I'd need a more formal proof) I'd expect the optimizer to have $U_Q = V_C$, as intuitively it makes sense that the determinant will be maximized when $Q$ is maximally aligned with $C$ in terms of singular vectors (with a similar logic to the Von Neumann trace inequality). Then the objective would look like:
${\rm det}\left( I +  \Sigma_C  \Sigma_Q  \Sigma_C \right) = \prod_i^N \left(1+\sigma_i(C)^2 \cdot \sigma_i(Q)\right)$, 
and I guess you could easily find a closed form solution for that under the constraint of $trace(Q) == 1$. But as you can see, the optimizer will depend on the singular values of $C=B^{-1/2}A$, so it won't be straightforward to relate the values of $B$ to the singular values of $C$, at least without assuming anything about $A$. 
Is there anything else you can assume about $A$?
