How to deal with the optimization problem with non-convex feasible set?

Question

How to deal with the problem with non-convex feasible set. \begin{align} &({\rm P1})\ \underset{\bf x}{\min} f({\bf x})={\bf x}^T{\bf x}+{\bf f}^T{\bf x}\\ &{s.t.}\\ &\|{\bf x}-{\bf x}_z\|_2\geq r,z=1,\cdots,M.\\ &{\bf 1}\leq{\bf x}\leq{\bf 10}, \end{align} where $r,M, {\bf x}_z,z=1,\cdots,M$ and $\bf f$ are constant. I have considered to relax the feasible set to be a convex one to employ SCA by using the first-order Taylor expansion at the solution of the $i$-th iteration ${\bf x}^i$ which is $\|{\bf x}-{\bf x}_z\|_2\geq\frac{1}{\|{\bf x^i}-{\bf x}_z\|_2}({\bf x^i}-{\bf x}_z)^T({\bf x}-{\bf x}_z)$. Therefore, $(\rm P1)$ can be transformed into \begin{align} &({\rm P2})\ \underset{\bf x}{\min} f({\bf x})={\bf x}^T{\bf x}+{\bf f}^T{\bf x}\\ &{s.t.}\\ &\frac{1}{\|{\bf x^i}-{\bf x}_z\|_2}({\bf x^i}-{\bf x}_z)^T({\bf x}-{\bf x}_z)\geq r,z=1,\cdots,M.\\ &{\bf 1}\leq{\bf x}\leq{\bf 10}. \end{align} Although $(\rm P2)$ is convex optimization problem, but the feasible set can be empty since the approximation greatly reduce the area of each original non-convex set, and the solution cannot be updated anymore. I'll really appreciate it if you can give me some insights to deal with the non-convex feasible set with specific structure.