Suppose I want to solve the following problem:
\begin{align*} min_{P, V}\ \log(|PVP^T|) + \text{tr}((PVP^T)^{-1}PQP^T) \end{align*} such that $PS = I$, where $S$ is a known matrix of full column rank and $Q$ is a positive definite matrix.
However, I don't really want to precisely estimate $V$, it is sufficient to estimate $PVP^T$.
Can I solve the above problem letting $R = PVP^T$ so that the objective function becomes $$log|R| + tr(R^{-1}PQP^T)$$ which is convex.
This leads $R = PQP^T$. Then plugging this solution back into the objective function, can I solve for $P$?
If so how can I ensure that the minimum has reached?