How can I perform {convex}^2? (Moderator note: {convex}^2 is convex, but the problem contained herein is not)

Nonconvex
1

Hi everyone,
please help me solve this problem.
My target function is like the following
max -(log2(1+A^2+B^2) - log2(1+B^2))

where A and B are complex cvx variables.

I already tried pow_p(x,p), but it doesn’t work.
I have no idea how to approximate this target function to make it work.

2

Can you first explain what you mean by maximizing the logarithm of a complex number.

Even if x is real \log(1+x^2) is neither convex nor concave so it is not looking good.

3

Thank you for your prompt reply, and I apologize for my delayed response.

My optimization problem is very similar to the one outlined in reference [1],
utilizing majorize minimization based algorithm,
and make some approximation to the constraints to convexify the problem.

However, the challenge lies in their problem where A^2 and B^2 can be set as real convex variables.
In my case, I need to set A and B as complex convex variables,
which is why I’m seeking guidance on how to perform {convex}^2.

[1] S.-H. Park, H. Lee, and S.-E. Hong, “Rate-splitting multiple access with conjugate beamforming for cell-free MIMO,” 2022 13th International Conference on Information and Communication Technology Convergence (ICTC), Oct. 2022.

4

Even if you use complex variable, in order to use CVX, the optimization problem must ultimately be reducible to a convex optimization problem in real variables. You have yet to provide a valid explanation as to why there is an underlying convex optimization problem in your case.