Some questions about log_det function,

as shown in the picture, i don not know why is error occurs, and i have test that the value of log_det(X) is different from log(det(X)), can you help me to solve these problems? thank you

Did you use the “optimal” variable value of a previous CVX problem instance to this problem? if so, the previous problem could have been infeasible, unbounded, or encountered numerical difficulty, so that the variable values were NaN after CVX completed. This resulted in NaN input to a subsequent CVX problem instance, which CVX considers to be invalid, hence the error message.

As for log)det(X) being different than log(det(X)), please show us the details supporting your claim.

It is not q good idea to use quiet until you have verified everything is working correctly, which evidently it is not yet.

help log_det

log_det Logarithm of the determinant of an SDP matrix.
For a square matrix X, log_det(X) returns
if X is symmetric (real) or Hermitian (complex) and positive semidefinite,
and -Inf otherwise.

When used in a CVX model, log_det(X) causes CVX's successive 
approximation method to be invoked, producing results exact to within
the tolerance of the solver. Therefore, whenever possible, the use of
DET_ROOTN(X) is to be preferred, because the latter choice can be
represented exactly in an SDP. For example, the objective
can be (and should be) replaced with
in fact, log_det(X) is implemented simply as N*LOG(DET_ROOTN(X)).

Disciplined convex programming information:
    log_det is concave and nonmonotonic; therefore, when used in
    CVX specifications, its argument must be affine.

thank you maybe the matrix X is not positive matrix

Does log_det(X) have the value -Inf for some numerical matrix X?

yes, the result is inf +a*i ,a is a finite number