My problem is that

\text{min}_{L}\,\, Tr(\Sigma(\gamma I - L)^{-1}) + \eta\parallel L \parallel_F^2

\text{s.t.}\,\, L \in \mathcal{L}, \gamma I \succeq L

And my question is that , how to express (\gamma I - L)^{-1} in CVX, because matrix power is not legal in CVX.

Let me presume that Sigma is psd, Let `C = chol(Sigma)`

.

Then

`trace(Sigma*inv(gama*eye(n)-L)) = trace(C'*C*inv(gama*eye(n)-L)) = trace(C*inv(gama*eye(n)-L)*C')`

by cyclic permutation invariance of trace.

Now apply my solution approach in Generalizing "trace_inv" for matrix quadratic forms .

Alternatively, use `Sigma = sqrtm(Sigma)*sqrtm(Sigma)`

instead of the Cholesky factorization, which will also handle the case in which the Cholesky factorization does not exist due to (near) singularity.

There are a few other comments by me at How to compute inverse of matrix .