Show that the Huber-loss based optimization is equivalent to $\ell_1$ norm based.
Solution 1:
The idea is much simpler. Use the fact that $$\min_{\mathbf{x}, \mathbf{z}} f(\mathbf{x}, \mathbf{z}) = \min_{\mathbf{x}} \left\{ \min_{\mathbf{z}} f(\mathbf{x}, \mathbf{z}) \right\}.$$ In your case, the solution of the inner minimization problem is exactly the Huber function.