Abstract
The Sylvester or smallest enclosing circle problem involves finding the smallest circle enclosing a finite number of points in the plane. We consider generalized versions of the Sylvester problem in which the points are replaced by sets. Based on the log-exponential smoothing technique and Nesterov’s accelerated gradient method, we present an effective numerical algorithm for solving these problems.
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Acknowledgments
The authors would like to thank the referees and Prof. Jie Sun for reading the paper carefully and giving valuable comments to improve the content and the presentation of the paper. The research of the first author was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant #101.01-2014.37. The research of Daniel Giles was partially supported by the USA National Science Foundation under grant DMS-1411817. The research of Nguyen Mau Nam was partially supported by the USA National Science Foundation under grant DMS-1411817 and the Simons Foundation under Grant #208785.
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Communicated by Horst Martini.
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An, N.T., Giles, D., Nam, N.M. et al. The Log-Exponential Smoothing Technique and Nesterov’s Accelerated Gradient Method for Generalized Sylvester Problems. J Optim Theory Appl 168, 559–583 (2016). https://doi.org/10.1007/s10957-015-0811-z
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DOI: https://doi.org/10.1007/s10957-015-0811-z
Keywords
- Log-exponential smoothing technique
- Majorization minimization algorithm
- Nesterov’s accelerated gradient method
- Generalized Sylvester problem