I am working on a problem of optimization of wireless sensor networks in order to achieve the optimal power allocation in the network. I find an equality matrix problem that can be written as follows: $\sum\limits_{i = 1}^m {{{(Q + A - {u_i}u_i^H)}^{ - 1}}{u_i}u_i^H{{(Q + A - {u_i}u_i^H)}^{ - 1}} = QBQ}$

$A$ and $B$ are given $PSD$ symmetric matrix. $U_i$ are also given and they verify: $\sum\limits_{i = 1}^m {{u_i}u_i^H = A +aI} $ $a$ is positive integer.

I'm looking for method to find the objective matrix$ (n,n)$ $Q \ge 0 $ "closed form or numeric solution of th PSD matrix $ Q$".