I modeled an optimization problem in an integer programming format. The main constraint I came up with is now nonconvex. I would like to see if there is another equivalent formulation in which the constraints are convex or simple enough to be solved.
\begin{equation} \begin{aligned} \min_{x,t} \quad & \sum_{i=1}^{n}{c_i x_i}\\ \textrm{s.t.} \quad & \sum_{j=1}^{n}{t_j \big( \text{dist} (v_i , v_j) - T x_j \big)} \leq 0 ,~ i=1,\dots,n \\ & t_1 + t_2 +\dots + t_n = 1\\ & t_i , x_i \in \{ 0 , 1 \} \\ \end{aligned} \end{equation}
Note that $\text{dist} (v_i , v_j) $ and $T$ are positive constants. $x_i = 1$ if the node $v_i$ is selected, $x_i = 0$ otherwise. $t_i$s are auxiliary variables. The constraints have been written in a way that for each node $v_i$, there exists a selected node $v_j$ such that $\text{dist} (v_i , v_j) \leq T.$