Skip to main content
SpringerLink
Log in

On the Complexity of a Mehrotra-Type Predictor-Corrector Algorithm

  • Conference paper
Computational Science and Its Applications – ICCSA 2012 (ICCSA 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7335))

Included in the following conference series:

Abstract

Based on the good computational results of the feasible version of the Mehrotra’s predictor-corrector variant algorithm presented by Bastos and Paixão, in this paper we discuss its complexity. We prove the efficiency of this algorithm by showing its polynomial complexity and, consequently, its Q-linearly convergence.

We start by proving some technical results which are used to discuss the step size estimate of the algorithm.

It is shown that, at each iteration, the step size computed by this Mehrotra’s predictor-corrector variant algorithm is bounded below, for n ≥ 2, by \(\frac{1}{200 n^4};\) consequently proving that the algorithm has O(n 4 |log(ε)|) iteration complexity.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Almeida, R., Bastos, F., Teixeira, A.: On polinomiality of a predictor-corrector variant algorithm. In: International Conference on Numerical Analysis and Applied Mathematica 2010 - ICNAAM 2010. AIP Conference Proceedings, vol. 2, pp. 959–963. Springer, New York (2010)

    Google Scholar 

  2. Bastos, F.: Problemas de transporte e métodos de ponto interior, Dissertação de Doutoramento, Universidade Nova de Lisboa, Lisboa (1994)

    Google Scholar 

  3. Bastos, F., Paixão, J.: Interior-point approaches to the transportation and assignment problems on microcomputers. Investigação Operacional 13(1), 3–15 (1993)

    Google Scholar 

  4. Freund, R.W., Jarre, F.: A QMR-based interior-point algorithm for solving linear programs, Numerical Analysis Manuscript, 94-19. AT&T Bell Laboratories, Murray Hill, N.J. (1994)

    Google Scholar 

  5. Güler, O., Ye, Y.: Convergence behavior of interior-point algorithms. Mathematical Programming 60, 215–228 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Karmarkar, N.K.: A new polinomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kojima, M., Megiddo, N., Mizuno, S.: A primal-dual infeasible-interior point algorithm for linear programming. Mathematical Programming, Series A 61, 261–280 (1993)

    MathSciNet  Google Scholar 

  8. Lustig, I.J., Marsten, R.E., Shanno, D.F.: Computational experience with a primal-dual interior point method for linear programming. Linear Algebra and Its Applications 152, 191–222 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lustig, I.J., Marsten, R.E., Shanno, D.F.: On implementing Mehrotra’s predictor-corrector interior-point method for linear programming. SIAM Journal on Optimization 2(3), 435–449 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lustig, I.J., Marsten, R.E., Shanno, D.F.: Interior point method for linear programming: Computational state of the art. ORSA Journal on Computing 6, 1–14 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. McShane, K.A., Monma, C.L., Shanno, D.F.: An implementation of a primal-dual interior point method for linear programming. ORSA Journal on Computing 1, 70–83 (1989)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Salahi, M., Peng, J., Terlaky, T.: On Mehrotra-Type Predictor-Corrector Algorithms. SIAM Journal on Optimization 18(4), 1377–1397 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  15. Ye, Y.: Interior Point Algorithms, Theory and Analysis. John Wiley and Sons, Chichester (1997)

    Book  MATH  Google Scholar 

  16. Zhang, Y., Zhang, D.: On polynomiality of the Mehrotra-type predictor-corrector interior-point algorithms. Mathematical Programming 68, 303–318 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhang, Y., Tapia, R.A., Dennis, J.E.: On the superlinear and quadratic convergence of primal-dual interior point linear programming algorithms. SIAM J. Optimization 2, 304–324 (1992)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Trás-os-Montes e Alto Douro, P-5000-911, Vila Real, Portugal

    Ana Paula Teixeira & Regina Almeida

  2. CIO. Faculty of Sciences, University of Lisbon, Portugal

    Ana Paula Teixeira

  3. CIDMA, University of Aveiro, Portugal

    Regina Almeida

Authors
  1. Ana Paula Teixeira
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Regina Almeida
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratory of Urban and Territorial Systems, University of Basilicata, 10, Viale dell’Ateneo Lucano, 85100, Potenza, Italy

    Beniamino Murgante

  2. Department of Mathematics and Computer Science, University of Perugia, Via Vanvitelli 1, 06123, Perugia, Italy

    Osvaldo Gervasi

  3. Department of Cyber Security Science, Federal University of Technology, Gidan Kwano Campus, Minna, Nigeria

    Sanjay Misra

  4. Faculty of Engineering, Department of Electronics Engineering and Telecommunications, State University of Rio de Janeiro, Rua Sao Francisco Xavier, 524, 50. andar, sala 5145-F, Maracana, 20.550-013, Rio de Janeiro, RJ, Brazil

    Nadia Nedjah

  5. Department of Production and Systems, University of Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Ana Maria A. C. Rocha

  6. School of Business Systems, Monash University, 3800, Clayton, VIC, Australia

    David Taniar

  7. Department of Intelligent Informatics, Kyushu Sangyo University, 2-3-1 Matsukadai, Higashi-ku, 813-8503, Fukuoka, Japan

    Bernady O. Apduhan

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Teixeira, A.P., Almeida, R. (2012). On the Complexity of a Mehrotra-Type Predictor-Corrector Algorithm. In: Murgante, B., et al. Computational Science and Its Applications – ICCSA 2012. ICCSA 2012. Lecture Notes in Computer Science, vol 7335. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31137-6_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-31137-6_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31136-9

  • Online ISBN: 978-3-642-31137-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation