Skip to main content
SpringerLink
Log in

New method for determining search directions for interior-point algorithms in linear optimization

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

We introduce an interior-point algorithm for linear optimization, which is based on a new technique for determining search directions. This method consists of a new type of transformation on the centering equations of the system which characterizes the central path. It can be shown that these new directions cannot be derived from usual kernel functions. Therefore, we extend the concept of the kernel functions, and we establish an equivalence between this approach and the proposed method for obtaining search directions. Moreover, we prove the polynomial complexity of the algorithm.

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

Fig. 1

Similar content being viewed by others

References

  1. Achache, M.: A new primal–dual path-following method for convex quadratic programming. Comput. Appl. Math. 25(1), 97–110 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Achache, M.: Complexity analysis and numerical implementation of a short-step primal–dual algorithm for linear complementarity problems. Appl. Math. Comput. 216(7), 1889–1895 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Ai, W., Zhang, S.: An O\((\sqrt{n} {L})\) iteration primal–dual path-following method, based on wide neighborhoods and large updates, for monotone LCP. SIAM J. Optim. 16(2), 400–417 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Chapter  Google Scholar 

  5. Asadi, S., Mansouri, H.: Polynomial interior-point algorithm for \({P}_*(\kappa )\) horizontal linear complementarity problems. Numer. Algorithms 63(2), 385–398 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bai, Y., El Ghami, M., Roos, C.: A comparative study of kernel functions for primal–dual interior-point algorithms in linear optimization. SIAM J. Optim. 15(1), 101–128 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bai, Y.Q., El Ghami, M., Roos, C.: A new efficient large-update primal–dual interior-point method based on a finite barrier. SIAM J. Optim. 13, 766–782 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dantzig, G.: Linear Programming and Extension. Princeton University Press, Princeton, NJ (1963)

    Book  MATH  Google Scholar 

  9. Darvay, Zs: A new algorithm for solving self-dual linear optimization problems. Studia Univ. Babeş-Bolyai Ser. Inf. 47(1), 15–26 (2002)

    MathSciNet  MATH  Google Scholar 

  10. Darvay, Zs: New interior point algorithms in linear programming. Adv. Model. Optim. 5(1), 51–92 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Darvay, Zs, Papp, I.M., Takács, P.R.: Complexity analysis of a full-Newton step interior-point method for linear optimization. Period. Math. Hung. 73(1), 27–42 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gay, D.: Electronic mail distribution of linear programming test problems. Math. Program. Soc. COAL Newsl. 3, 10–12 (1985)

    Google Scholar 

  13. Gondzio, J., Terlaky, T.: A computational view of interior point methods for linear programming. In: Beasley, J. (ed.) Advances in Linear and Integer Programming. Oxford University Press, Oxford (1995)

    Google Scholar 

  14. Illés, T., Terlaky, T.: Pivot versus interior point methods: pros and cons. Eur. J. Oper. Res. 140, 6–26 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Illés, T., Nagy, M., Terlaky, T.: EP theorem for dual linear complementarity problems. J. Optim. Theory Appl. 140(2), 233–238 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Illés, T., Nagy, M., Terlaky, T.: Polynomial interior point algorithms for general linear complementarity problems. Algoritm. Oper. Res. 5(1), 1–12 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Illés, T., Nagy, M., Terlaky, T.: A polynomial path-following interior point algorithm for general linear complementarity problems. J. Global Optim. 47(3), 329–342 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jansen, B., Roos, C., Terlaky, T.: The theory of linear programming: skew symmetric self-dual problems and the central path. Optimization 29, 225–233 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Kheirfam, B.: A predictor-corrector interior-point algorithm for \({P}_*(\kappa )\)-horizontal linear complementarity problem. Numer. Algorithms 66(2), 349–361 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Klafszky, E., Terlaky, T.: Some generalizations of the criss-cross method for the linear complementarity problem of oriented matroids. Combinatorica 9(2), 189–198 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems. Lecture Notes in Computer Science, vol. 538. Springer Verlag, Berlin, Germany (1991)

  23. Li, Y., Terlaky, T.: A new class of large neighborhood path-following interior point algorithms for semidefinite optimization with \({O}(\sqrt{n}\log \frac{\rm {T}r(x^0s^0)}{\epsilon })\) iteration complexity. SIAM J. Optim. 20(6), 2853–2875 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mansouri, H., Pirhaji, M.: A polynomial interior-point algorithm for monotone linear complementarity problems. J. Optim. Theory Appl. 157(2), 451–461 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mansouri, H., Siyavash, T., Zangiabadi, M.: A path-following infeasible interior-point algorithm for semidefinite programming. Iran. J. Oper. Res. 3(1), 11–30 (2012)

    Google Scholar 

  26. 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 

  27. Nesterov, Y.E., Nemirovskii, A.S.: Interior Point Polynomial Methods in Convex Programming, SIAM Studies in Applied Mathematics, vol. 13. SIAM Publications, Philadelphia (1994)

    MATH  Google Scholar 

  28. Peng, J., Roos, C., Terlaky, T.: Self-Regular Functions: A New Paradigm for primal–dual Interior-Point Methods. Princeton University Press, Princeton (2002)

    MATH  Google Scholar 

  29. Peyghami, M., Hafshejani, S.: Complexity analysis of an interior-point algorithm for linear optimization based on a new proximity function. Numer. Algorithms 67(1), 33–48 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Potra, F.: A superlinearly convergent predictor-corrector method for degenerate LCP in a wide neighborhood of the central path with O\(\left(\sqrt{n}L\right)\) iteration complexity. Math. Program. 100(2), 317–337 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Potra, F.A., Sheng, R.: Predictor-corrector algorithm for solving \({P}_*(\kappa )\)-matrix LCP from arbitrary positive starting points. Math. Program. 76(1), 223–244 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Potra, F.A., Sheng, R.: A large-step infeasible-interior-point method for the \({P}^*\)-Matrix LCP. SIAM J. Optim. 7(2), 318–335 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Roos, C., Terlaky, T., Vial, J-Ph: Theory and Algorithms for Linear Optimization. Springer, New York (2005)

    MATH  Google Scholar 

  34. Sonnevend, G.: An “analytic center” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. In: Prékopa, A., Szelezsán, J., Strazicky, B. (eds.) System Modelling and Optimization: Proceedings of the 12th IFIP-Conference held in Budapest, Hungary, September 1985. Lecture Notes in Control and Information Sciences, vol. 84, pp. 866–876. Springer Verlag, Berlin, West-Germany (1986)

  35. Terlaky, T.: A convergent criss-cross method. Optimization 16(5), 683–690 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  36. Terlaky, T.: An easy way to teach interior-point methods. Eur. J. Oper. Res. 130(1), 1–19 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, G.Q.: A new polynomial interior-point algorithm for the monotone linear complementarity problem over symmetric cones with full NT-steps. Asia-Pacific J. Oper. Res. 29(2), 1250,015 (2012)

    Article  MathSciNet  Google Scholar 

  38. Wang, G.Q., Bai, Y.Q.: A new primal–dual path-following interior-point algorithm for semidefinite optimization. J. Math. Anal. Appl. 353(1), 339–349 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang, G.Q., Bai, Y.Q.: A primal–dual interior-point algorithm for second-order cone optimization with full Nesterov-Todd step. Appl. Math. Comput. 215(3), 1047–1061 (2009)

    MathSciNet  MATH  Google Scholar 

  40. Wang, G.Q., Bai, Y.Q.: A new full Nesterov–Todd step primal–dual path-following interior-point algorithm for symmetric optimization. J. Optim. Theory Appl. 154(3), 966–985 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Zhang, L., Xu, Y.: A full-Newton step interior-point algorithm based on modified Newton direction. Oper. Res. Lett. 39, 318–322 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang, M., Bai, Y., Wang, G.: A new primal–dual path-following interior-point algorithm for linearly constrained convex optimization. J. Shanghai Univ. 12(6), 475–480 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are thankful for the valuable suggestions of the editor and the reviewers that helped improve the clarity and overall presentation of the paper. The authors are also grateful for the support of Edutus College (Collegium Talentum). This work was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project Number PN-III-P4-ID-PCE-2016-0190.

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, No. 1 Mihail Kogălniceanu Street, 400084, Cluj-Napoca, Romania

    Zsolt Darvay & Petra-Renáta Takács

Authors
  1. Zsolt Darvay
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petra-Renáta Takács
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zsolt Darvay.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Darvay, Z., Takács, PR. New method for determining search directions for interior-point algorithms in linear optimization. Optim Lett 12, 1099–1116 (2018). https://doi.org/10.1007/s11590-017-1171-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-017-1171-4

Keywords

Search

Navigation