Part of my problem is maximizing the minimum magnitude squared of vector elements (x = [x_i]). This is not a convex problem as min is taking convex expressions.

One way to work with it is by maximizing g = log_prod(d(x)), where d=[d_i] and d_i = |x_i|^2. log_prod is concave and nonincreasing, and d is convex of x. Such composition is concave, so maximizing g is a convex problem. However, CVX is not accepting such formulation.

Why is it difficult for CVX to discover such concavity? does it help to build an atom for g?

"Such composition is concave, "

How did you conclude that? I am not aware of any such composition rule. Please provide a proof in accordance with the FAQ.

Thanks for reply.

Assume f(x) = h(g(x)), then f is concave if h is concave and nonincreasing and g is convex (see Convex Optimization by Boyd and Vandenberghe, section 3.24).

In our case h == log_prod. As mentioned in the referecen manual of CVX, log_prod is concave and non increasing. Also, we have g == d, which is convex of x as it takes the absolute square element-wise.

Yes, I am well aware of that book and section. There is no such composition rule listed there.