I am currently working on a mathematical problem involving a non-convex function, $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$. I have a constrained optimization problem that I would like to convert into an unconstrained problem. The objective is to find the center and radius of the minimum enclosing ball of the non-convex set defined by the function $f(x)$.
The constrained problem is formulated as follows:
\begin{align*} \max_{\textbf{z} \in \mathbb{R}^{d}, r } \quad & r\\ \textrm{s.t.} \quad & f(\textbf{x}) \leq 0 \,\,\, \forall \textbf{x}, \lVert{\textbf{x}- \textbf{z}\rVert}_{2} = r \\ &r \geq0 \ \end{align*}
My intuition is that this should be well-studied problem. Any references or how to solve above would be useful