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.

翻译：线性规划中的单纯形法在实践中表现出极高的效率，从理论角度理解其性能一直是活跃的研究课题。为此目的，Spielman和Teng (JACM '04) 首次引入的平滑分析框架，将求解具有 $d$ 个变量和 $n$ 个约束的线性规划的平滑复杂度定义为：当向线性规划数据添加方差为 $\sigma^2$ 的高斯噪声时，算法运行时间的期望值。我们证明了单纯形法的平滑复杂度为 $O(\sigma^{-3/2} d^{13/4}\log^{7/4} n)$，相较于先前 $O(\sigma^{-2} d^2\sqrt{\log n})$ 的界，改进了对 $1/\sigma$ 的依赖关系。我们通过对早期分析中的关键——\emph{影子界}——进行新的分析实现了这一改进。为了展示我们新方法的效力，我们使用该方法证明了二维多边形平滑复杂度的近乎紧的上界。我们还首次建立了单纯形法平滑复杂度的非平凡下界，证明了\emph{影子顶点单纯形法}以高概率至少需要 $\Omega \Big(\min \big(\sigma^{-1/2} d^{-1/2}\log^{-1/4} d,2^d \big) \Big)$ 个转轴步骤。我们分析的一个关键部分是对正 $2^k$ 边形扩展形式的一种新变体。最后，我们通过一个数值实验表明，该分析有可能得到进一步改进。