Formulation of an SDP problem


I am a new forum member and CVX user. After reading the CVX user guide, the forum’s FAQ, and searching for similar questions, I decided to post this question.

I have formed the following convex optimization problem using the SDP with rank-1 relaxation approach:

Next, I have to model it in CVX. As a new user of CVX, I found difficulties. If I try to model the problem exactly as it is presented here, I get DCP error in the first constraint (8b): Cannot perform the operation: {positive constant} ./ {real affine}. It happens all the time as I can see in previous posts. I have to note that there is no inf or NaN in the employed scalars, vectors, or matrices (although the vectors \mathbf{h}_{k} have small numeric values, in the order of 10^{-3}\times (a+jb). Searching for a solution in the forum, I found out that the use of inv_pos instead of division might be the answer to my problem. However, that was not the problem.

Maybe is something simple that I overlooked. Could you guide me towards a workaround?


You are implying that the denominator of the LHS of (8b) is positive. So how about multiplying both sides of (8b) by that denominator; that moves the denominator to the RHS. Replace N/\rho with N*inv_pos(\rho). That results in affine >= convex, which CVX will accept.

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I struggled with the details and I didn’t see the obvious solution… This worked as a charm, as expected! A million thanks!