How to solve such a log_det problem using cvx

1

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)MMAT(:,:,reali)’*AT(:,:,reali)MM’AR(:,:,reali)’);
subject to
abs(trace(MMMM’))<=NtNr;
cvx_end
The cvx_solver I use is Mosek.

2

The constraint can be reformulated as
norrm(M,'fro') <= sqrt(Nt^Nr)

Have you proven the objective is convex? If so, perhaps you need to use a determinantal inequality?

3

Thank you for your advice, Mark! The objective function is convex because the matrix in \mathrm{det}() is positive-semidefinite.

What’s more, I have tried your advice. The error comes from the expression of objective function, which says:“Disciplined convex programming error:
Only scalar quadratic forms can be specified in CVX”.

4

This is not the determinant of an affine semidefinite argument.

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