Problem: Let us consider the following minimization problem \begin{align*} &\min_{(x,t) \in \mathbb{R}^n\times \mathbb{R}} t\\ \text{s.t }& \Vert x-q_i\Vert^2 \le r_i^2 + t,\ \forall i = \overline{1,k}, \end{align*} where $q_i$ and $r_i$ are known value. Show that the problem admits an optimal value.
My attempt: I intend to show that the problem admits a minimizer by using Weierstrass's theorem. To do that, the most important thing is to show the constraint set is bounded. Easily, we can see that $t$ is bounded from below. Now, if $t$ is bounded from above, everything is done (since it yields $x$ is bounded also). So, I wonder if we can assume for free the upper bound of $t$?