Packing of Circles on Square Flat Torus as Global Optimization of Mixed Integer Nonlinear problem
| Autor: | Smirnov, Sergey A., Voloshinov, Vladimir V. |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 (B&B). 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 B&B-solver SCIP, http://scip.zib.de, and its parallel implementation ParaSCIP, http://ug.zib.de, had been used in computing experiments to find "very good" approximations of optimal arrangements. The main result is a confirmation of the conjecture on optimal packing for N=9 that was published in 2012 by O. Musin and A. Nikitenko. To do that, ParaSCIP took about 2000 CPU*hours (16 hours x 128 CPUs) of cluster HPC4/HPC5, National Research Centre "Kurchatov Institute", http://ckp.nrcki.ru Comment: 9 pages, 8 figures, 2 tables |
| Databáze: | arXiv |
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