Advertisement

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

  • Murat MutEmail author
  • Tamás Terlaky
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification

52B05 65K05 68Q25 90C05 90C51 90C60 

Notes

Acknowledgments

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.

References

  1. 1.
    Sonnevend, G., Stoer, J., Zhao, G.: On the complexity of following the central path of linear programs by linear extrapolation II. Math. Progr. 52, 527–553 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Zhao, G., Stoer, J.: Estimating the complexity of a class of path-following methods for solving linear programs by curvature integrals. Appl. Math. Optim. 27, 85–103 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Deza, A., Terlaky, T., Zinchenko, Y.: A continuous \(d\)-step conjecture for polytopes. Discrete Comput. Geom. 41, 318–327 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Klee, V., Walkup, D.W.: The \(d\)-step conjecture for polyhedra of dimension \(d<6\). Acta Math. 117, 53–78 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Zhao, G.: Representing the space of linear programs as the Grassmann manifold. Math. Progr. 121, 353–386 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Monteiro, R.D., Tsuchiya, T.: 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. Math. Progr. 115, 105–149 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Vavasis, S.A., Ye, Y.: A primal-dual interior point method whose running time depends only on the constraint matrix. Math. Progr. 74, 79–120 (1996)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Roos, C., Terlaky, T., Vial, J.P.: Interior Point Methods for Linear Optimization. Springer, Berlin (2005)zbMATHGoogle Scholar
  9. 9.
    Deza, A., Terlaky, T., Xie, F., Zinchenko, Y.: Diameter and curvature: intriguing analogies. Electron. Notes Discrete Math. 31, 221–225 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Deza, A., Terlaky, T., Zinchenko, Y.: Polytopes and arrangements: diameter and curvature. Oper. Res. Lett. 36(2), 215–222 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations