Packing of Circles on Square Flat Torus as Global Optimization of Mixed Integer Nonlinear problem

The article demonstrates rather general approach to problems of discrete geometry: treat them as global optimization problems to be solved by one of general purpose solver implementing branch-and-bound algorithm. This approach may be used for various types of problems, i.e. Tammes problems, Thomson problems, search of minimal potential energy of micro-clusters, etc. Here we consider a problem of densest packing of equal circles in special geometrical object, so called square flat torus $\mathbb{R}^2/\mathbb{Z}^2$ with the induced metric. It is formulated as Mixed-Integer Nonlinear Problem with linear and non-convex quadratic constraints. The open-source solver SCIP, this http URL, and its parallel implementation ParaSCIP, this http URL, had been used in computing experiments. The main result is "computer aided" proof of the conjecture on optimal packing for N=9 that was published in 2012. To do that, ParaSCIP took about 2059 CPU*hours (16 hours x 128 CPUs) of cluster HPC4/HPC5, National Research Centre "Kurchatov Institute".