Upper and Lower Bounds on the Smoothed Complexity of the Simplex Method
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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 σ^2 is added to the LP data. We prove that the smoothed complexity of the simplex method is O(σ^-3/2 d^13/4log^7/4 n), improving the dependence on 1/σ compared to the previous bound of O(σ^-2 d^2√(log n)). 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. We also establish the first non-trivial lower bound on the smoothed complexity of the simplex method, proving that the shadow vertex simplex method requires at least Ω(min(σ^-1/2 d^-1/2log^-1/4 d,2^d ) ) 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.
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