I have the following optimization problem:
\begin{align*} \min\;&f(x)\\ \mathrm{s.t.}\;&\rho^\top \mathbf{1}=1\\ & \rho^\top g+w^\top G\leq y\\ &w^\top w+\rho^\top \rho-z^2\leq 0\\ &z\geq 0\\ &w =\rho \circ x \end{align*}
where $g$ and $G$ are given constants. I have a few more constraints incorporating $x,z$ and $y$, which are convex. My issue is the last constraint which is bilinear. I wonder is there a way to convexify this problem? I have tried using McCormick relaxation but the resulting constraint is too loose.