I am trying to solve the following optimization problem

f(X) = y'*X*y - logdet(X) + r'*X^{-1}r.

where X is symmetric and positive definite. I couldn’t add the X^{-1} term in CVX. Is there any way to add it in the objective function. I couldn’t see it in the manual. After following instructions of Bien, I tried the following. However, it showed some errors. Further I found this matrix_frac term that lets me add the r'X^{-1}r term directly

> cvx_begin sdp
> variable X(N,N) hermitian
> variable t
> minimize(.5*y'*X*y - 0.5*log_det(X) + t)
> subject to
> X >= 0
> [t, r'; r, X] >= 0
> cvx_end
Successive approximation method to be employed.
SDPT3 will be called several times to refine the solution.
Original size: 1618 variables, 1088 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 8.000e+00 6.975e+00 0.000e+00 | Failed
1/ 1 | 3.457e+00 6.535e-01 0.000e+00 | Failed
0/ 0 | 0.000e+00 0.000e+00 0.000e+00 | Failed
0/ 0 | 0.000e+00 0.000e+00 0.000e+00 | Failed
0/ 0 | 0.000e+00 0.000e+00 0.000e+00 | Failed
-----------------------------------------------------------------
Status: Failed
Optimal value (cvx_optval): NaN

Successive approximation method to be employed.
SDPT3 will be called several times to refine the solution.
Original size: 1618 variables, 1088 equality constraints
1 exponentials add 8 variables, 5 equality constraints