An Analogue of the Klee–Walkup Result for Sonnevend’s Curvature of the Central Path

  • Murat MutEmail author
  • Tamás Terlaky


For linear optimization problems, we consider a curvature integral of the central path which was first introduced by Sonnevend et al. (Math Progr 52:527–553, 1991). Our main result states that in order to establish an upper bound for the total Sonnevend curvature of the central path of a polytope, it is sufficient to consider only the case where the number of inequalities is twice as many as the number of variables. This also implies that the worst cases of linear optimization problems for certain path-following interior-point algorithms can be reconstructed for the aforementioned case. As a by-product, our construction yields an asymptotically worst-case lower bound in the order of magnitude of the number of inequalities for Sonnevend’s curvature. Our research is motivated by the work of Deza et al. (Electron Notes Discrete Math 31:221–225, 2008) for the geometric curvature of the central path, which is analogous to the Klee–Walkup result for the diameter of a polytope.


Curvature Central path Polytopes Diameter  Complexity Interior-point methods Linear optimization 

Mathematics Subject Classification

52B05 65K05 68Q25 90C05 90C51 90C60 



Research was supported by a Start-up grant of Lehigh University. It was also supported by TAMOP-4.2.2.A-11/1KONV-2012-0012: Basic research for the development of hybrid and electric vehicles. The TAMOP Project was supported by the European Union and co-financed by the European Regional Development Fund.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Nielsen Marketing AnalyticsEvanstonUSA
  2. 2.Department of Industrial and Systems EngineeringLehigh UniversityBethlehemUSA

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