Skip to main content
Log in

Further development of multiple centrality correctors for interior point methods

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

This paper addresses the role of centrality in the implementation of interior point methods. We provide theoretical arguments to justify the use of a symmetric neighbourhood, and translate them into computational practice leading to a new insight into the role of re-centering in the implementation of interior point methods. Second-order correctors, such as Mehrotra’s predictor–corrector, can occasionally fail: we derive a remedy to such difficulties from a new interpretation of multiple centrality correctors. Through extensive numerical experience we show that the proposed centrality correcting scheme leads to noteworthy savings over second-order predictor–corrector technique and previous implementations of multiple centrality correctors.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Andersen, E.D., Gondzio, J., Mészáros, C., Xu, X.: Implementation of interior point methods for large scale linear programming. In: Terlaky, T. (ed.) Interior Point Methods in Mathematical Programming, pp. 189–252. Kluwer Academic, Dordrecht (1996)

    Google Scholar 

  2. Carpenter, T.J., Lustig, I.J., Mulvey, J.M., Shanno, D.F.: Higher-order predictor-corrector interior point methods with application to quadratic objectives. SIAM J. Optim. 3, 696–725 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cartis, C.: Some disadvantages of a Mehrotra-type primal–dual corrector interior-point algorithm for linear programming. Technical report 04/27, Numerical Analysis Group, Computing Laboratory, Oxford University (2004)

  4. Cartis, C.: On the convergence of a primal–dual second-order corrector interior point algorithm for linear programming. Technical report 05/04, Numerical Analysis Group, Computing Laboratory, Oxford University (2005)

  5. Czyzyk, J., Mehrotra, S., Wright, S.: PCx user guide. Technical report OTC 96/01, Optimization Technology Center (May 1996)

  6. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gondzio, J.: HOPDM (version 2.12)—a fast LP solver based on a primal–dual interior point method. Eur. J. Oper. Res. 85, 221–225 (1995)

    Article  MATH  Google Scholar 

  8. Gondzio, J.: Multiple centrality corrections in a primal–dual method for linear programming. Comput. Optim. Appl. 6, 137–156 (1996)

    MATH  MathSciNet  Google Scholar 

  9. Haeberly, J.-P., Nayakkankuppam, M., Overton, M.: Extending Mehrotra and Gondzio higher order methods to mixed semidefinite-quadratic–linear programming. Optim. Methods Softw. 11, 67–90 (1999)

    Article  MathSciNet  Google Scholar 

  10. Jarre, F., Wechs, M.: Extending Merhotra’s corrector for linear programs. Adv. Model. Optim. 1, 38–60 (1999)

    MATH  Google Scholar 

  11. Lustig, I.J., Marsten, R.E., Shanno, D.F.: On implementing Mehrotra’s predictor–corrector interior-point method for linear programming. SIAM J. Optim. 2, 435–449 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Mehrotra, S.: On the implementation of a primal–dual interior point method. SIAM J. Optim. 2, 575–601 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mehrotra, S., Li, Z.: Convergence conditions and Krylov subspace-based corrections for primal–dual interior-point method. SIAM J. Optim. 15, 635–653 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Mizuno, S., Todd, M., Ye, Y.: On adaptive step primal–dual interior-point algorithms for linear programming. Math. Oper. Res. 18, 964–981 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  16. Salahi, M., Peng, J., Terlaky, T.: On Mehrotra-type predictor–corrector algorithms. AdvOl report 2005/4, McMaster University (2005)

  17. Tapia, R., Zhang, Y., Saltzman, M., Weiser, A.: The Mehrotra predictor–corrector interior-point method as a perturbed composite Newton method. SIAM J. Optim. 6, 47–56 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Vavasis, S.A., Ye, Y.: A primal–dual interior point method whose running time depends only on the constraint matrix. Math. Program. 74, 79–120 (1996)

    MathSciNet  Google Scholar 

  19. Wright, S.J.: Primal–Dual Interior-Point Methods. SIAM, Philadelphia (1997)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marco Colombo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Colombo, M., Gondzio, J. Further development of multiple centrality correctors for interior point methods. Comput Optim Appl 41, 277–305 (2008). https://doi.org/10.1007/s10589-007-9106-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-007-9106-0

Keywords

Navigation