Can CVX handle the function log_det( I + inv(X) )

4

I believe
t \geq \log(\det( I + {\rm inv}(X) ))

is equivalent to
-t \leq \log(\det( I - {\rm inv}(X+I) ))

is equivalent to
-t \leq \log(\det( I - U ))
U \succeq {\rm inv}(X + I)

is equivalent to
-t \leq \log(\det( I - U ))
\left[\begin{array}{cc}X+I & I\\I & U\end{array}\right] \succeq 0

The first step follows from the scalar to matrix generalization above which you can prove more rigidly. The second step follows by applying \log(\det( I - U )) \leq \log(\det( I - V )) on I \succeq U \succeq V to our case with V = {\rm inv}(X + I). The last step is the Schur complement lemma.

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