Mathematical Programming

, Volume 115, Issue 1, pp 105–149 | Cite as

A strong bound on the integral of the central path curvature and its relationship with the iteration-complexity of primal-dual path-following LP algorithms

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Abstract

The main goals of this paper are to: i) relate two iteration-complexity bounds derived for the Mizuno-Todd-Ye predictor-corrector (MTY P-C) algorithm for linear programming (LP), and; ii) study the geometrical structure of the LP central path. The first iteration-complexity bound for the MTY P-C algorithm considered in this paper is expressed in terms of the integral of a certain curvature function over the traversed portion of the central path. The second iteration-complexity bound, derived recently by the authors using the notion of crossover events introduced by Vavasis and Ye, is expressed in terms of a scale-invariant condition number associated with m × n constraint matrix of the LP. In this paper, we establish a relationship between these bounds by showing that the first one can be majorized by the second one. We also establish a geometric result about the central path which gives a rigorous justification based on the curvature of the central path of a claim made by Vavasis and Ye, in view of the behavior of their layered least squares path following LP method, that the central path consists of \({\mathcal{O}}(n^2)\) long but straight continuous parts while the remaining curved part is relatively “short”.

Keywords

Interior-point algorithms Primal-dual algorithms Path-following Central path Layered least squares steps Condition number Polynomial complexity Crossover events Scale-invariance Predictor-corrector Affine scaling Strongly polynomial Linear programming Curvature 

Mathematics Subject Classification

65K05 68Q25 90C05 90C51 90C60 
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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA
  2. 2.The Institute of Statistical MathematicsTokyoJapan

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