Projection on a convex set that defined by variables not showing in the objective

Question

The projection problem I want to solve is to find a vector $u\in R^d$ so that $\min_{(u,z)}: \|| u'-u\||^2_2$, where $u_i\leq z_i, z \in \Omega$. Here $\Omega$ is a convex set. As shown all $z_i$ are not in the objective.

Can we solve the problem as follows:

1. Solve $\min_{(u,z)}: \|| u'-u\||^2_2$, where $u_i\leq z_i$, and find $(u^*, z^*)$.
2. If $z^* \notin \Omega$, we can find $z$ such that $\min_z \||z^*-z\||_2^2$, where $z\in \Omega$.
3. Repeat 1 and 2 until converge.

Because it is very efficient to solve $\min_z \||z^*-z\||_2^2$, where $z\in \Omega$, I want to solve the original problem by taking advantage of it. Any thoughts on how to use this for a fast algorithm?

Thank you so much.