Skip to main content
SpringerLink
Log in

A Predictor-corrector algorithm with multiple corrections for convex quadratic programming

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

Abstract

Recently an infeasible interior-point algorithm for linear programming (LP) was presented by Liu and Sun. By using similar predictor steps, we give a (feasible) predictor-corrector algorithm for convex quadratic programming (QP). We introduce a (scaled) proximity measure and a dynamical forcing factor (centering parameter). The latter is used to force the duality gap to decrease. The algorithm can decrease the duality gap monotonically. Polynomial complexity can be proved and the result coincides with the best one for LP, namely, \(O(\sqrt{n}\log n\mu^{0}/\varepsilon)\).

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andersen, E.D., Ye, Y.: A computational study of the homogeneous algorithm for large-scale convex optimization. Comput. Optim. Appl. 10(3), 243–269 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cafieri, S., D’Apuzzo, M., Marino, M., Mucherino, A., Toraldo, G.: Interior-point solver for large-scale quadratic programming problems with bound constraints. J. Optim. Theory Appl. 129(1), 55–75 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Colombo, M., Gondzio, J.: Further development of multiple centrality correctors for interior point methods. Comput. Optim. Appl. 41(3), 277–305 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dominguez, J., González-Lima, M.D.: A primal-dual interior-point algorithm for quadratic programming. Numer. Algorithms 42(1), 1–30 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gondzio, J.: Multiple centrality corrections in a primal-dual method for linear programming. Comput. Optim. Appl. 6(2), 137–156 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kapoor, S., Vaidya, P.M.: Fast algorithms for convex quadratic programming and multicommodity flows. In: Proceedings of the 18th Annual ACM Symposium on Theory of Computing, California, 1986, pp. 147–159 (1986)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Kojima, M., Megiddo, N., Mizuno, S.: A primal-dual infeasible-interior-point algorithm for linear programming. Math. Program. 61(3), 263–280 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, Z., Sun, W.: An infeasible interior-point algorithm with full-Newton step for linear optimization. Numer. Algorithms 46(2), 173–188 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Lu, Z., Monteiro, R.D.C., O’Neal, J.W.: An iterative solver-based infeasible primal-dual path-following algorithm for convex quadratic programming. SIAM J. Optim. 17(1), 287–310 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mansouri, H., Roos, C.: Simplified O(nL) infeasible interior-point algorithm for linear optimization using full-Newton step. Optim. Methods Softw. 22(3), 519–530 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mehrotra, S.: On the implementation of a primal-dual interior point method. SIAM J. Optim. 2(4), 575–601 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mizuno, S., Todd, M.J., Ye, Y.: On adaptive step primal-dual interior-point algorithms for linear programming. Math. Oper. Res. 18(4), 964–981 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Monteiro, R.D.C., Adler, I.: Interior path following primal-dual algorithms, Part II: convex quadratic programming. Math. Program. (1989)

  15. Peng, J., Terlaky, T., Zhao, Y.: A predictor-corrector algorithm for linear optimization based on a specific self-regular proximity function. SIAM J. Optim. 15(4), 1105–1127 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Roos, C., Terlaky, T., Vial, J.-Ph.: Theory and Algorithms for Linear Optimization. An Interior Approach. Wiley, Chichester (1997)

    MATH  Google Scholar 

  17. Wu, F., Wu, S., Ye, Y.: On quadratic convergence of the \(O(\sqrt{n}L)\)-iteration homogeneous and self-dual linear programming algorithm. Ann. Oper. Res. 87(0), 393–406 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ye, Y., Tse, E.: A polynomial algorithm for convex programming. Working Paper, Department of Engineering-Economic Systems, Stanford University, Stanford, CA (1986)

  19. Ye, Y., Todd, M.J., Mizuno, S.: An \(O(\sqrt{n})L\)-iteration homogeneous and self-dual linear programming algorithm. Math. Oper. Res. 19(1), 53–67 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Science, Hohai University, Nanjing, 210098, China

    Zhongyi Liu

  2. Jincheng College, Nanjing University of Aeronautics and Astronautics, Nanjing, 211156, China

    Yue Chen

  3. School of Mathematical Science, Nanjing Normal University, Nanjing, 210097, China

    Wenyu Sun

  4. School of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing, 210094, China

    Zhihui Wei

Authors
  1. Zhongyi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yue Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wenyu Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhihui Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhongyi Liu.

Additional information

This work was supported by the Fundamental Research Funds for the Central Universities of China (No. 2009B27314) and also by the National Natural Science Foundation of China (No. 61071146, No. 10871098).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, Z., Chen, Y., Sun, W. et al. A Predictor-corrector algorithm with multiple corrections for convex quadratic programming. Comput Optim Appl 52, 373–391 (2012). https://doi.org/10.1007/s10589-011-9421-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-011-9421-3

Keywords

Search

Navigation