Upper and Lower Bounds on the Smoothed Complexity of the Simplex Method
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| Publicado en: | arXiv.org (May 15, 2024), p. n/a |
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| Otros Autores: | , |
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Cornell University Library, arXiv.org
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| Acceso en línea: | Citation/Abstract Full text outside of ProQuest |
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| 022 | |a 2331-8422 | ||
| 035 | |a 2739286587 | ||
| 045 | 0 | |b d20240515 | |
| 100 | 1 | |a Huiberts, Sophie | |
| 245 | 1 | |a Upper and Lower Bounds on the Smoothed Complexity of the Simplex Method | |
| 260 | |b Cornell University Library, arXiv.org |c May 15, 2024 | ||
| 513 | |a Working Paper | ||
| 520 | 3 | |a The simplex method for linear programming is known to be highly efficient in practice, and understanding its performance from a theoretical perspective is an active research topic. The framework of smoothed analysis, first introduced by Spielman and Teng (JACM '04) for this purpose, defines the smoothed complexity of solving a linear program with \(d\) variables and \(n\) constraints as the expected running time when Gaussian noise of variance \(\sigma^2\) is added to the LP data. We prove that the smoothed complexity of the simplex method is \(O(\sigma^{-3/2} d^{13/4}\log^{7/4} n)\), improving the dependence on \(1/\sigma\) compared to the previous bound of \(O(\sigma^{-2} d^2\sqrt{\log n})\). We accomplish this through a new analysis of the \emph{shadow bound}, key to earlier analyses as well. Illustrating the power of our new method, we use our method to prove a nearly tight upper bound on the smoothed complexity of two-dimensional polygons. We also establish the first non-trivial lower bound on the smoothed complexity of the simplex method, proving that the \emph{shadow vertex simplex method} requires at least \(\Omega \Big(\min \big(\sigma^{-1/2} d^{-1/2}\log^{-1/4} d,2^d \big) \Big)\) pivot steps with high probability. A key part of our analysis is a new variation on the extended formulation for the regular \(2^k\)-gon. We end with a numerical experiment that suggests this analysis could be further improved. | |
| 653 | |a Lower bounds | ||
| 653 | |a Linear programming | ||
| 653 | |a Random noise | ||
| 653 | |a Complexity | ||
| 653 | |a Simplex method | ||
| 653 | |a Upper bounds | ||
| 653 | |a Shadows | ||
| 700 | 1 | |a Lee, Yin Tat | |
| 700 | 1 | |a Zhang, Xinzhi | |
| 773 | 0 | |t arXiv.org |g (May 15, 2024), p. n/a | |
| 786 | 0 | |d ProQuest |t Engineering Database | |
| 856 | 4 | 1 | |3 Citation/Abstract |u https://www.proquest.com/docview/2739286587/abstract/embedded/7BTGNMKEMPT1V9Z2?source=fedsrch |
| 856 | 4 | 0 | |3 Full text outside of ProQuest |u http://arxiv.org/abs/2211.11860 |