Packing of Circles on Square Flat Torus as Global Optimization of Mixed Integer Nonlinear problem
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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 R^2/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, http://scip.zib.de, and its parallel implementation ParaSCIP, http://ug.zib.de, 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".
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