Upper and Lower Bounds on the Smoothed Complexity of the Simplex Method

TL;DR

This work provides the first nontrivial upper and lower bounds for the smoothed complexity of the simplex method under Gaussian perturbations, advancing beyond prior σ-dependent estimates. By developing a refined shadow-bound analysis and a novel edge-counting strategy, the authors prove an upper bound of $\mathcal{S}(n,d,\sigma)=O(\sigma^{-3/2} d^{13/4} \log^{5/4} n)$ for small $\sigma$, improving the dependence on $1/\sigma$ relative to prior results. They also establish a lower bound, showing $\mathcal{S}(4d-13,d,\sigma)=\Omega\big(\min(1/(\sqrt{d\sigma\sqrt{\log d}}), 2^d)\big)$, using an extended-formulation-inspired construction for a perturbed 2D shadow, and validate the bounds with a two-dimensional polygon analysis and numerical experiments. The results imply that smoothed complexity can be polynomial in $n$, $d$, and $\sigma^{-1}$ with improved exponents, and they illuminate the geometric structure of the shadow under perturbations. Overall, the paper strengthens the theoretical understanding of why simplex methods perform well in practice under small random perturbations and provides tools potentially applicable to related geometric-optimization problems.

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 $σ^2$ is added to the LP data. We prove that the smoothed complexity of the simplex method is $O(σ^{-3/2} d^{13/4}\log^{7/4} n)$, improving the dependence on $1/σ$ compared to the previous bound of $O(σ^{-2} d^2\sqrt{\log n})$. 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 $Ω\Big(\min \big(σ^{-1/2} d^{-1/2}\log^{-1/4} d,2^d \big) \Big)$ 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.

Figures (2)

• Figure 4: Illustration of the case when $e_{I^+}$ is short. In purple is the edge $e_I$ with its extension line dashed. The next edge in clockwise direction, $e_I^+$, has length $\ell_{I^+}$ and is drawn in black. In red is the line-to-vertex distance $\operatorname{dist}(p_{I^+}, \operatorname{affhull}(e_I))$, and in blue is the angle $\theta_I$. If $\operatorname{dist}(p_{I^+}, \operatorname{affhull}(e_I)) \geq \gamma$ then $\theta_I \geq \sin(\theta_I) \geq \gamma / \ell_{I^+}$.
• Figure 7: Measured shadow sizes for sampled perturbations of our construction, for different values of $k$ and $\sigma$.

Theorems & Definitions (50)

• Theorem 1.1: Smoothed Complexity of Shadow Vertex Simplex Algorithm
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Two-Dimensional Upper Bound
• definition 2.1: Polytope
• definition 2.2: Valid Condition and Facet
• definition 2.3: Polar dual of a convex body
• definition 2.7: Gaussian distribution
• Lemma 2.8: Gaussian tail bound
• corollary 2.9: Global diameter of Gaussian random variables