SIAM Journal on Computing, Volume 49, Issue 5, Page STOC18-449-STOC18-499, January 2020.
Explaining the excellent practical performance of the simplex method for linear programming has been a major topic of research for over 50 years. One of the most successful frameworks for understanding the simplex method was given by Spielman and Teng [J. ACM, 51 (2004), pp. 385--463] who developed the notion of smoothed analysis. Starting from an arbitrary linear program (LP) with $d$ variables and $n$ constraints, Spielman and Teng analyzed the expected runtime over random perturbations of the LP, known as the smoothed LP, where variance $\sigma^2$ Gaussian noise is added to the LP data. In particular, they gave a two-stage shadow vertex simplex algorithm which uses an expected $\widetilde{O}(d^{55} n^{86} \sigma^{-30} + d^{70}n^{86})$ number of simplex pivots to solve the smoothed LP. Their analysis and runtime was substantially improved by Deshpande and Spielman [FOCS `05, 2005, pp. 349--356] and later Vershynin [SIAM J. Comput., 39 (2009), pp. 646--678]. The fastest current algorithm, due to Vershynin, solves the smoothed LP using an expected $O\big(\log^2 n \cdot \log\log n \cdot (d^3\sigma^{-4} + d^5\log^2 n + d^9\log^4 d)\big)$ number of pivots, improving the dependence on $n$ from polynomial to polylogarithmic. While the original proof of Spielman and Teng has now been substantially simplified, the resulting analyses are still quite long and complex and the parameter dependencies far from optimal. In this work, we make substantial progress on this front, providing an improved and simpler analysis of shadow simplex methods, where our algorithm requires an expected $O(d^2 \sqrt{\log n} ~ \sigma^{-2} + d^3 \log^{3/2} n)$ number of simplex pivots. We obtain our results via an improved shadow bound, key to earlier analyses as well, combined with improvements on algorithmic techniques of Vershynin. As an added bonus, our analysis is completely modular and applies to a range of perturbations, which, aside from Gaussians, also includes Laplace perturbations.

中文翻译：

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单纯形法的友好平滑分析

SIAM计算杂志，第49卷，第5期，第STOC18-449-STOC18-499页，2020年1月。
五十多年来，解释用于线性编程的单纯形法的出色实用性能一直是研究的主要课题。Spielman和Teng提出了最简单的理解单纯形法的框架之一。ACM，51（2004），第385--463页]提出了平滑分析的概念。Spielman和Teng从具有$ d $变量和$ n $约束的任意线性程序（LP）开始，分析了LP的随机扰动（称为平滑LP）的预期运行时间，其中方差$ \ sigma ^ 2 $高斯噪声已添加到LP数据。特别是，他们给出了一个两阶段阴影顶点单纯形算法，该算法使用了预期的$ \ widetilde {O}（d ^ {55} n ^ {86} \ sigma ^ {-30} + d ^ {70} n ^ { 86}）$个单纯形枢轴的数目，以求解平滑的LP。Deshpande和Spielman [FOCS`05，2005，第349--356页]和后来的Vershynin [SIAM J. Comput。，39（2009），第646--678页]大大改善了他们的分析和运行时间。由于Vershynin，目前最快的算法使用期望的$ O \ big（\ log ^ 2 n \ cdot \ log \ log n \ cdot（d ^ 3 \ sigma ^ {-4} + d ^ 5 \ log ^ 2 n + d ^ 9 \ log ^ 4 d）\ big）$枢轴数，改善了对$ n $从多项式到多对数的依赖性。虽然现在已经大大简化了Spielman和Teng的原始证明，但所得的分析仍然相当漫长和复杂，并且参数依赖性远非最佳。在这项工作中，我们在这方面取得了实质性进展，提供了对阴影单纯形方法的改进和更简单的分析，其中我们的算法需要预期的$ O（d ^ 2 \ sqrt {\ log n}〜\ sigma ^ {-2} + d ^ 3 \ log ^ {3/2} n）$单纯形枢轴数。我们通过改进阴影界限（也是早期分析的关键）以及对Vershynin算法技术的改进来获得结果。另外，我们的分析是完全模块化的，适用于一系列扰动，除了高斯算子外，还包括拉普拉斯扰动。