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View all- García-Pacheco F(2023)On Bishop–Phelps and Krein–Milman PropertiesMathematics10.3390/math1121447311:21(4473)Online publication date: 28-Oct-2023
Upper and Lower Bounds on the Smoothed Complexity of the Simplex Method
Pages 1904 - 1917
Abstract
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 σ2 is added to the LP data. We prove that the smoothed complexity of the simplex method is O(σ−3/2 d13/4log7/4 n), improving the dependence on 1/σ compared to the previous bound of O(σ−2 d2√logn). We accomplish this through a new analysis of the 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.
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Index Terms
- Upper and Lower Bounds on the Smoothed Complexity of the Simplex Method
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View all- García-Pacheco F(2023)On Bishop–Phelps and Krein–Milman PropertiesMathematics10.3390/math1121447311:21(4473)Online publication date: 28-Oct-2023
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