I have a set of service locations $\{ {\bf{x}}_i \in \mathbb{R}^2\}_{i = 1}^{N}$.
I have $k$ facilities to locate in a way that distance of each service point to the nearest facility is minimized.
My model is building "circles" around the facilities and minimize the radius of the circle while constraining that each service location is within at least one facility radius:
$$ \min \| {\bf{r}} \|_{\infty} $$
subject to
$$ \forall i \min_k \| {\bf{x}}_{i} - {\bf{c}}_{k} \| \leq {r}_{k} $$
Where $\bf{r} \in \mathbb{R}^k$ is the vector of all $k$ radii of around the facilities and ${\bf{c}}_{k}$ is the location of the $k$ facility. The should mean that each "service point" is within the radius of its nearest facility.
The formulation above isn't convex as those choice of ${r}_{k}$ is a function of the left side.
Is there a way to convexify this? Or other similar in the spirit convex solution?
The idea is building a service radius around each facility.