I have such a convex optimization problem:

\begin{aligned}
\mathcal{P}_{3}: \min _{\mathbf{X}} &-\log _{2} \operatorname{det}\left(\mathbf{I}_{N_{r}}+\frac{S N R}{N_{r}} \mathbf{A}^{\mathrm{H}}\left(\mathbf{X} \otimes \mathbf{I}_{N_{t}}\right) \mathbf{A}\right) \\
\text { s.t. } & \operatorname{Tr}\left(\mathbf{A}^{\mathrm{H}}\left(\mathbf{X} \otimes \mathbf{I}_{N_{t}}\right) \mathbf{A}\right)-N_{r} N_{t} \leq 0 \\
&-\mathbf{X}_{i, j} \leq 0 \\
& \mathbf{X}=\mathbf{X}^{\mathrm{T}}
\end{aligned}

where \mathbf{X} is a positive semipositive matrix. The convexity of the problem is proved which is actually a SDP. * When I use CVX, the optimal solution has negative elements so it does not satisfy the second constraint.* The cvx_status is ‘Solved’ and the solver I used is Mosek. My cvx code is followed as:

cvx_begin sdp quiet

variable pattern_cov(L,L) symmetric;

minimize -det_rootn(eye(Nr)+snr/min(Nt,Nr)*A(:,:,reali)’*kron(pattern_cov,eye(Nt))*A(:,:,reali));

subject to

pattern_cov>=0;

abs(trace(A(:,:,reali)’*kron(pattern_cov,eye(Nt))

*A(:,:,reali)))<=Nt*Nr;

abs(trace(A(:,:,reali)’*kron(pattern_cov.’,eye(Nt))

*A(:,:,reali)))<=Nt*Nr;

pattern_cov==semidefinite(L);

cvx_end

How can I solve this problem?