How to solve generalized SDP in CVX?


I want to solve the following optimization problem.

$$P=\min \log \det (P^{-1})$$
$$[\text{s.t. }
A^TP+PA & PB \
B^TP & -c \
\leq 0]$$ where P \in R^{n,n} and A \in R^{n,n}, B\in R^{n}, c\in R are known.

The hard constraint can be written as an LMI in in P_{i,j}. Semidefine program (SDP) also have LMI constraint but with linear objective in P_{i,j}. So, this problem can be considered as a generalization of SDP. Can anyone point me a solver for this?


det(invĀ§) = 1/detĀ§. So how about