Little example. Here, we try to pack \(n\) circles with a given radius \(r_i\) into a larger disc with an unknown radius \(R\). The goal is to minimize \(R\). The underlying model is simple:
| Packing of Circles |
|---|
| \[\begin{align} \min\> & \color{darkred}R \\ & \sum_c \left(\color{darkred}p_{i,c}-\color{darkred}p_{j,c}\right)^2 \ge \left(\color{darkblue}r_i+\color{darkblue}r_j\right)^2 & \forall i\lt j \\ & \sum_c \color{darkred}p_{i,c}^2 \le \left(\color{darkred}R-\color{darkblue}r_i\right)^2 & \forall i \\ & \color{darkred}R \ge 0\\ & c \in \{x,y\} \\ \end{align}\] |
