Smoothed Analysis of Renegar's Condition Number for Linear Programming

For any linear program, we show that a slight random relative perturbation of that linear program has small condition number with high probability. Following [ST01], we call this smoothed analysis of the condition number. Part of our main result is that the expectation of the log of the condition number of any appropriately scaled linear program subject to a Gaussian perturbation of variance σ is at most O(lognd/σ) with high probability. Since the condition number bounds the running time of many algorithms for convex programming, this may explain their observed fast convergence.