Advertisement

Periodica Mathematica Hungarica

, Volume 73, Issue 1, pp 27–42 | Cite as

Complexity analysis of a full-Newton step interior-point method for linear optimization

  • Zsolt DarvayEmail author
  • Ingrid-Magdolna Papp
  • Petra-Renáta Takács
Article

Abstract

This paper concerns a short-update primal-dual interior-point method for linear optimization based on a new search direction. We apply a vector-valued function generated by a univariate function on the nonlinear equation of the system which defines the central path. The common way to obtain the equivalent form of the central path is using the square root function. In this paper we consider a new function formed by the difference of the identity map and the square root function. We apply Newton’s method in order to get the new directions. In spite of the fact that the analysis is more difficult in this case, we prove that the complexity of the algorithm is identical with the one of the best known methods for linear optimization.

Keywords

Linear optimization Interior-point method Full-Newton step Search direction Polynomial complexity 

Mathematics Subject Classification

90C05 90C51 

Notes

Acknowledgments

The authors gratefully acknowledge the support of the Edutus College (Collegium Talentum) and the Transylvanian Museum Society. The authors are thankful to the editor and the reviewer for their helpful suggestions that improved the presentation of this paper. We also thank Ágnes Felméri and Nóra Forró for their useful comments on the paper. The work of the third author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0024.

References

  1. 1.
    M. Achache, A weighted-path-following method for the linear complementarity problem. Studia Univ. Babeş-Bolyai. Ser. Informatica 49(1), 61–73 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    M. Achache, A new primal-dual path-following method for convex quadratic programming. Comput. Appl. Math. 25(1), 97–110 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    M. Achache, Complexity analysis and numerical implementation of a short-step primal-dual algorithm for linear complementarity problems. Appl. Math. Comput. 216(7), 1889–1895 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    K. Ahmadi, F. Hasani, B. Kheirfam, A full-Newton step infeasible interior-point algorithm based on Darvay directions for linear optimization. J. Math. Model. Algorithms Oper. Res. 13(2), 191–208 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    W. Ai, S. Zhang, 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    F. Alizadeh, D. Goldfarb, Second-order cone programming. Math. Program. 95(1), 3–51 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    E. Andersen, J. Gondzio, Cs Mészáros, X. Xu, Implementation of interior point methods for large scale linear programs, in Inter. Point Methods Math. Program., ed. by T. Terlaky (Kluwer Academic Publishers, Dordrecht, 1996), pp. 189–252CrossRefGoogle Scholar
  8. 8.
    S. Asadi, H. Mansouri, Polynomial interior-point algorithm for \({P}_*(\kappa )\) horizontal linear complementarity problems. Numer. Algorithms 63(2), 385–398 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    S. Asadi, H. Mansouri, A new full-Newton step \({O}(n)\) infeasible interior-point algorithm for \({P}_*(\kappa )\)-horizontal linear complementarity problems. Comput. Sci. J. Moldova 22(1), 37–61 (2014)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Y.Q. Bai, L.M. Sun, Y. Chen, A new path-following interior-point algorithm for monotone semidefinite linear complementarity problems. Dyn. Contin. Discrete Impuls. Syst., Ser. B: Appl. Algorithms 17, 769–783 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Y.Q. Bai, F.Y. Wang, X.W. Luo, A polynomial-time interior-point algorithm for convex quadratic semidefinite optimization. RAIRO-Oper. Res. 44(3), 251–265 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    R.W. Cottle, J.S. Pang, R.E. Stone, The Linear Complementarity Problem. Computer Science and Scientific Computing (Academic Press, Boston, 1992)Google Scholar
  13. 13.
    Zs. Darvay, A new algorithm for solving self-dual linear optimization problems. Studia Univ. Babeş-Bolyai, Ser. Informatica 47(1), 15–26 (2002)Google Scholar
  14. 14.
    Zs. Darvay, A weighted-path-following method for linear optimization. Studia Univ. Babeş-Bolyai, Ser. Informatica 47(2), 3–12 (2002)Google Scholar
  15. 15.
    Zs. Darvay, New interior point algorithms in linear programming. Adv. Model. Optim. 5(1), 51–92 (2003)Google Scholar
  16. 16.
    Zs. Darvay, Á. Felméri, N. Forró, I.-M. Papp, P.-R. Takács, A new interior-point algorithm for solving linear optimization problems, in XVII. FMTU, ed. by E. Bitay (Transylvanian Museum Society, Cluj-Napoca, 2012), pp. 87–90. In HungarianGoogle Scholar
  17. 17.
    Zs. Darvay, I.-M. Papp, P.-R. Takács, An infeasible full-Newton step algorithm for linear optimization with one centering step in major iteration. Studia Univ. Babeş-Bolyai, Ser. Informatica 59(1), 28–45 (2014)Google Scholar
  18. 18.
    J. Gondzio, T. Terlaky, A computational view of interior point methods for linear programming, in Advances in Linear and Integer Programming, ed. by J. Beasley (Oxford University Press, Oxford, GB, 1995)Google Scholar
  19. 19.
    M. Halická, Analytical properties of the central path at boundary point in linear programming. Math. Program. 84(2), 335–355 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    M. Halická, Two simple proofs for analyticity of the central path in linear programming. Oper. Res. Lett. 28(1), 9–19 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    T. Illés, Lineáris optimalizálás elmélete és belsőpontos algoritmusai. Operations Research Reports 2014-04, Eötvös Loránd University, Budapest, pp. 1–95 (2014)Google Scholar
  22. 22.
    T. Illés, M. Nagy, A Mizuno-Todd-Ye type predictor-corrector algorithm for sufficient linear complementarity problems. Eur. J. Oper. Res. 181(3), 1097–1111 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    T. Illés, M. Nagy, T. Terlaky, EP theorem for dual linear complementarity problems. J. Optim. Theory Appl. 140(2), 233–238 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    T. Illés, M. Nagy, T. Terlaky, Polynomial interior point algorithms for general linear complementarity problems. Alg. Oper. Res. 5(1), 1–12 (2010)MathSciNetzbMATHGoogle Scholar
  25. 25.
    T. Illés, M. Nagy, T. Terlaky, A polynomial path-following interior point algorithm for general linear complementarity problems. J. Global. Optim. 47(3), 329–342 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    T. Illés, T. Terlaky, Pivot versus interior point methods: Pros and Cons. Eur. J. Oper. Res. 140, 6–26 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    N. Karmarkar, A new polynomial-time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    B. Kheirfam, A new infeasible interior-point method based on Darvay’s technique for symmetric optimization. Ann. Oper. Res. 211(1), 209–224 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    B. Kheirfam, A predictor-corrector interior-point algorithm for \({P}_*(\kappa )\)-horizontal linear complementarity problem. Numer. Algorithms 66(2), 349–361 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    E.D. Klerk, Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications (Kluwer Academic Pubishers, Norwell, MA, 2002)CrossRefzbMATHGoogle Scholar
  31. 31.
    M. Kojima, N. Megiddo, T. Noma, A. Yoshise, A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, vol. 538, Lecture Notes in Computer Science (Springer, Berlin, 1991)zbMATHGoogle Scholar
  32. 32.
    M. Kojima, N. Megiddo, Y. Ye, An interior point potential reduction algorithm for the linear complementarity problem. Math. Program. 54(1–3), 267–279 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    M. Kojima, S. Mizuno, A. Yoshise, A polynomial-time algorithm for a class of linear complementarity problems. Math. Program. 44(1–3), 1–26 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    I. Lustig, R. Marsten, D. Shanno, On implementing Mehrotra’s predictor-corrector interior-point method for linear programming. SIAM J. Optim. 2(3), 435–449 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    I. Lustig, R. Marsten, D. Shanno, Computational experience with a globally convergent primal-dual predictor-corrector algorithm for linear programming. Math. Program. 66, 123–135 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    H. Mansouri, M. Pirhaji, A polynomial interior-point algorithm for monotone linear complementarity problems. J. Optim. Theory Appl. 157(2), 451–461 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    H. Mansouri, T. Siyavash, M. Zangiabadi, A path-following infeasible interior-point algorithm for semidefinite programming. Iran. J. Oper. Res. 3(1), 11–30 (2012)Google Scholar
  38. 38.
    S. Mehrotra, On the implementation of a primal-dual interior point method. SIAM J. Optim. 2(4), 575–601 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    C. Mészáros, The efficient implementation of interior point methods for linear programming and their applications. Ph.D. thesis, Eötvös Loránd University of Sciences, Ph.D. School of Operations Research, Applied Mathematics and Statistics, Budapest (1996)Google Scholar
  40. 40.
    R.D.C. Monteiro, Y. Zhang, A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite programming. Math. Program. 81(3), 281–299 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Y. Nesterov, A. Nemirovskii, Interior Point Polynomial Methods in Convex Programming: Theory and Algorithms, SIAM Studies in Applied Mathematics, vol. 13 (SIAM Publications, Philadelphia, 1994)CrossRefzbMATHGoogle Scholar
  42. 42.
    J. Peng, C. Roos, T. Terlaky, Self-Regular Functions: A New Paradigm for Primal-Dual Interior-Point Methods (Princeton University Press, Princeton, 2002)zbMATHGoogle Scholar
  43. 43.
    F.A. Potra, The Mizuno-Todd-Ye algorithm in a larger neighborhood of the central path. Eur. J. Oper. Res. 143(2), 257–267 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    F.A. Potra, R. Sheng, Predictor-corrector algorithm for solving \({P}_*(\kappa )\)-matrix LCP from arbitrary positive starting points. Math. Program. 76(1), 223–244 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    F.A. Potra, R. Sheng, A large-step infeasible-interior-point method for the \({P}^*\)-Matrix LCP. SIAM J. Optim. 7(2), 318–335 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    C. Roos, T. Terlaky, J-Ph Vial, Theory and Algorithms for Linear Optimization (Springer, New York, 2005)zbMATHGoogle Scholar
  47. 47.
    S. Schmieta, F. Alizadeh, Extension of primal-dual interior point algorithms to symmetric cones. Math. Program., Ser. A 96(3), 409–438 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    G. Sonnevend, An ”analytic center” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. in System Modelling and Optimization: Proceedings of the 12th IFIP-Conference held in Budapest, Hungary, September 1985, Lecture Notes in Control and Information Sciences. ed. by A Prékopa, J Szelezsán, B Strazicky, vol. 84 (Springer, Berlin, 1986) pp. 866–876Google Scholar
  49. 49.
    T. Terlaky, An easy way to teach interior-point methods. Eur. J. Oper. Res. 130(1), 1–19 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    M.V.C Vieira, Jordan algebraic approach to symmetric optimization. Ph.D. thesis, Electrical Engineering, Mathematics and Computer Science, Delft University of Technology (2007)Google Scholar
  51. 51.
    G.Q. Wang, A new polynomial interior-point algorithm for the monotone linear complementarity problem over symmetric cones with full NT-steps. Asia-Pac. J. Oper. Res. 29(2), 1250,015 (2012)MathSciNetCrossRefGoogle Scholar
  52. 52.
    G.Q. Wang, Y.Q. Bai, A new primal-dual path-following interior-point algorithm for semidefinite optimization. J. Math. Anal. Appl. 353(1), 339–349 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    G.Q. Wang, Y.Q. Bai, A primal-dual interior-point algorithm for second-order cone optimization with full Nesterov-Todd step. Appl. Math. Comput. 215(3), 1047–1061 (2009)MathSciNetzbMATHGoogle Scholar
  54. 54.
    G.Q. Wang, Y.Q. Bai, 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)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    G.Q. Wang, Y.J. Yue, X.Z. Cai, Weighted-path-following interior-point algorithm to monotone mixed linear complementarity problem. Fuzzy Inf. Eng. 1(4), 435–445 (2009)CrossRefzbMATHGoogle Scholar
  56. 56.
    G.Q. Wang, Y.J. Yue, X.Z. Cai, A weighted-path-following method for monotone horizontal linear complementarity problem. in Fuzzy Information and Engineering, Advances in Soft Computing. ed. by B.y. Cao, C.y. Zhang, T.f. Li, vol. 54, (Springer, Berlin, 2009) pp. 479–487Google Scholar
  57. 57.
    S. Wright, Primal-Dual Interior-Point Methods (SIAM, Philadelphia, 1997)CrossRefzbMATHGoogle Scholar
  58. 58.
    Y. Ye, Interior Point Algorithms. Theory and Analysis (Wiley, Chichester, 1997)CrossRefzbMATHGoogle Scholar
  59. 59.
    Y. Ye, A path to the Arrow-Debreu competitive market equilibrium. Math. Program. 111(1–2), 315–348 (2008)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Y. Ye, M. Todd, S. Mizuno, An \(O(\sqrt{n}L)\)-iteration homogeneous and self-dual linear programming algorithm. Math. Oper. Res. 19, 53–67 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2016

Authors and Affiliations

  • Zsolt Darvay
    • 1
    Email author
  • Ingrid-Magdolna Papp
    • 1
  • Petra-Renáta Takács
    • 1
  1. 1.Faculty of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania

Personalised recommendations