I have the following problem and would like to formulate that as an SDP. I am not sure how to approach this :
A set $S$ is given such that : $$ S = \{P \in R^{n \times m} : ||p_i - c_i|| \leq d_i \}$$ Here $c_i \in R^n$ is the i-th column of $C$ and $d_i \in R^+ $. Another full-column rank matrix $M$ is $ n \times m $
$P^TB + B^TP $ is PSD $ \forall \ \ P \in S$
Then I have to convert the following optimization problem into an SDP $$\min_P tr(P(I_m + P^TB + B^T)^{-1}P^T)$$ such that $$P \in S $$
How would I atleast approach this to convert into the following form :
$$ \min C \cdot X $$ such that $$ A_i \cdot X = b_i$$ $$ x \geq 0$$ Here, $ X \in R^{n \times n}$ , $A_i \in R^{n \times n}$ , $ b_i \in R $ and $C \in R^{n \times n} $