Assume you have matrices F, W*, P* and we have the following optimization problem over matrix P
$$ \bf{ min (|W^* F P^* | - |W^* FP |})$$
can we say the above problem is similar to
$$\bf{min |P^*|-|P|}$$
If no, then how can i simplify the first problem? thank you
I am assuming $W^*FP^*$ , $W^*FP$, $P^*$, and $P$ are all squares since we are talking about determinants. Suppose $P$ and $P^*$ are both in $\mathbb{R}^{n \times n}$, then $W^*F \in \mathbb{R}^{n \times n}$ as well.
Then we have $$|W^*FP^*|-|W^*FP|=|W^*F|(|P^*|-|P|)$$
It is possible that $|W^*F|$ is non-positive, hence the question need not be equivalent. It is equivalent when $|W^*F|>0$.