Minimizing the norm of a vector which is squared elementwise

Question

I've encountered the following optimization problem, amazingly in a real-world engineering application:

$\min ||\omega^2||_2$

$s.t. H\omega=M$

where $\omega^2=[\omega_1^2\quad\omega_2^2\quad\omega_3^2\quad ...]$.

I tried implementing this in CVXPY in a roundabout way by minimizing $||\Omega\omega||$, where $\Omega$ is a matrix with $\omega$ on it's diagonal, but the software said the problem was nonconvex. This wasn't a huge surprise to me (although I can't immediately see the nonconvexity).

Does anyone have any ideas about tackling this problem, either directly or through a simplification? One idea I had was the following:

1. Solve for the least squares solution $\omega_l=H^\dagger M$
2. Take the elementwise square root of the least squares solution, $v=\sqrt{\omega_l}$
3. Scale $v$ such that $Hv=M$ using the following: $\omega=\frac{||M||}{||Hv||}v$

Despite being completely unconstrained to satisfying $H\omega=M$, the method actually worked in several cases, but failed to be robust in all of the necessary scenarios. This problem has proven surprisingly fascinating, so I'd love any ideas people have!