where the variable v_k is ld matrix and tau is a scalar,however in this case,log_det(H v_k*v_k’ *H)and chol()can not be accepted by cvx ,and i try log_det(norm(v_k,2)) instead but it returns the following error:’'Disciplined convex programming error: Invalid computation: geo_mean( {convex} ),So,can someone give me some idea how to make it work by the cvx ,very grateful for your help and thanks a lot.

Thank you for your attention.
In fact,by adopting this transformation,the term log_det(W_k) become: \left | W_k \right |=\frac{1}{\left | J_{k} \right |}\begin{vmatrix}-J_k & V_k^{H}H^{H}\\ HV_k & I\end{vmatrix}
and the corresponding cvx code is something like:

which still rejected by the cvx,return the error:"Error using cvxprob/newcnstr (line 123)Both sides of an SDP constraint must be affine."
but thanks for your help anyway

where * stands for the remaining terms of T_k except \log |W_k|, hoping that cvx is able to handle the cost function. If all minima are negative, the original problem has a solution for the value of \tau.