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 (B&B). This approach may be used for various types of problems, i.e. Tammes problems...
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| creator | Smirnov, Sergey A Voloshinov, Vladimir V |
| description | 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 |
| doi_str_mv | 10.48550/arxiv.1809.10525 |
| format | Article |
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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</description><identifier>DOI: 10.48550/arxiv.1809.10525</identifier><language>eng</language><subject>Computer Science - Distributed, Parallel, and Cluster Computing ; Mathematics - Optimization and Control</subject><creationdate>2018-09</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,776,881</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/1809.10525$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.1809.10525$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Smirnov, Sergey A</creatorcontrib><creatorcontrib>Voloshinov, Vladimir V</creatorcontrib><title>Packing of Circles on Square Flat Torus as Global Optimization of Mixed Integer Nonlinear problem</title><description>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
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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</abstract><doi>10.48550/arxiv.1809.10525</doi><oa>free_for_read</oa></addata></record> |
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| source | arXiv.org |
| subjects | Computer Science - Distributed, Parallel, and Cluster Computing Mathematics - Optimization and Control |
| title | Packing of Circles on Square Flat Torus as Global Optimization of Mixed Integer Nonlinear problem |
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