I have such a convex optimization problem:

\begin{aligned}
&\min _{\boldsymbol{M}}-\log _{2} \operatorname{det}\left(I_{N_{r}}+\rho \boldsymbol{A}_{\mathrm{R}} \boldsymbol{M} \boldsymbol{A}_{\mathrm{T}}^{\mathrm{H}} \boldsymbol{A}_{\mathrm{T}} \boldsymbol{M}^{\mathrm{H}} \boldsymbol{A}_{\mathrm{R}}^{\mathrm{H}}\right) \\
&\text { s.t. } \operatorname{trace}\left( \boldsymbol{M} \boldsymbol{M}^{\mathrm{H}} \right) \leq N_{t} N_{r}
\end{aligned}

The I_{N_{r}} is an identity matrix with dimension N_{r}. A_{R},A_{T} are response matrix with dimension N_{r}\times L,N_{t}\times L,which both have independent columns. M is diagonal matrix with real number. The main point is the description of objective function. My cvx code now:

cvx_begin quiet

variable MM(L,L) diagonal;

minimize -det_rootn(eye(Nr)+SNR/min(Nt,Nr)*AR(:,:,reali)*MM*AT(:,:,reali)â€™*AT(:,:,reali)*MMâ€™ AR(:,:,reali)â€™);*Nr;

subject to

abs(trace(MMMMâ€™))<=Nt

cvx_end

The cvx_solver I use is Mosek.