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

Hi everyone,
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.

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.

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.

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.