i think the sum of two type 2 norm is still convex and for the variable phi ，it is a convex set。but now i’m not sure whether the relationship between the variable and the objective function is still convex。what if i set exp(j*phi) as the variable?will their relation become more simple? i really thanks for your guiding.
The sum of squared two-norms is convex (you never even get to that point, because the argument of norm is not affine). But you try to minimize the reciprocal of that, which is not convex.
And you have a two-norm squared on the left-hand side of >= constraint. That is non-convex. If the constraint really is convex, that will require consolidating the norm terms and being left on the RHS with a sum of convex terms.
There’s plenty more screwed up stuff with your program. There are some strange things in your program, for which I have no idea what the behavior of CVX is or “is supposed to” be. You define a variable, then an expression in terms of that variable (so far, that would be o.k,), but then you assign the variable in terms of the expression you assigned in terms of that variable (I don’t know how that works). I have no idea what phi(j) <In> rand(1)*2*pi does (apparently some kind of constraint is intended), but I doubt it does what you want.
Anyhow, I really didn’t want to get into all that, so I stated before, as I state again, that you need to show this problem is convex, which looks rather dubious to me, to put it mildly.
I don’t know whether it works “fine”, but there are no fatal errors. As for the objective function, please read the link I posted. I believe you will need to use something other than CVX for your problem.
Do you understand that maximizing a convex function is the same as minimizing a concave function, and that is not a convex optimization problem?