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MP

2011

2011

We perform a smoothed analysis of Renegar’s condition number for linear programming by analyzing the distribution of the distance to ill-posedness of a linear program subject to a slight Gaussian perturbation. In particular, we show that for every n-by-d matrix ¯A, n-vector ¯b, and d-vector ¯c satisfying ¯A, ¯b, ¯c F ≤ 1 and every σ ≤ 1, E A,b,c [log C(A, b, c)] = O(log(nd/σ)), where A, b and c are Gaussian perturbations of ¯A, ¯b and ¯c of variance σ2 and C(A, b, c) is the condition number of the linear program deﬁned by (A, b, c). From this bound, we obtain a smoothed analysis of interior point algorithms. By combining this with the smoothed analysis of ﬁnite termination of Spielman and Teng (Math. Prog. Ser. B, 2003), we show that the smoothed complexity of interior point algorithms for linear programming is O(n3 log(nd/σ)).

Related Content

Added |
16 May 2011 |

Updated |
16 May 2011 |

Type |
Journal |

Year |
2011 |

Where |
MP |

Authors |
John Dunagan, Daniel A. Spielman, Shang-Hua Teng |

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