Mathematical Programming

, Volume 126, Issue 2, pp 315–350 | Cite as

Smoothed analysis of condition numbers and complexity implications for linear programming

  • John Dunagan
  • Daniel A. Spielman
  • Shang-Hua Teng
Full Length Paper Series A


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 Ā, n-vector \({\bar{\varvec b}}\), and d-vector \({\bar{\varvec c}}\) satisfying \({{||\bar{\bf A}, \bar{\varvec b}, \bar{\varvec c}||_F \leq 1}}\) and every σ ≤ 1,
$$\mathop{\bf E}\limits_{\bf A,\varvec b,\varvec c }{{[\log C (\bf A,\varvec b,\varvec c)} = O (\log (nd/\sigma)),}$$
where A, b and c are Gaussian perturbations of Ā, \({\bar{\varvec b}}\) and \({\bar{\varvec c}}\) 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 (n3log(nd/σ)).

Mathematics Subject Classification (2000)

90C05 90C51 


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Copyright information

© Springer and Mathematical Programming Society 2009

Authors and Affiliations

  • John Dunagan
    • 1
  • Daniel A. Spielman
    • 2
  • Shang-Hua Teng
    • 3
  1. 1.Microsoft ResearchBellevueUSA
  2. 2.Department of Computer Science, Program in Applied MathematicsYale UniversityNew HavenUSA
  3. 3.Department of Computer ScienceBoston UniversityBostonUSA

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