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A New Complexity Analysis for Full-Newton Step Infeasible Interior-Point Algorithm for Horizontal Linear Complementarity Problems

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Abstract

In this paper, we first present a full-Newton step feasible interior-point algorithm for solving horizontal linear complementarity problems. We prove that the full-Newton step to the central path is quadratically convergent. Then, we generalize an infeasible interior-point method for linear optimization to horizontal linear complementarity problems based on new search directions. This algorithm starts from strictly feasible iterates on the central path of a perturbed problem that is produced by a suitable perturbation in the horizontal linear complementarity problem. We use the so-called feasibility steps that find strictly feasible iterates for the next perturbed problem. By using centering steps for the new perturbed problem, we obtain a strictly feasible iterate close enough to the central path of the new perturbed problem. The complexity of the algorithm coincides with the best known iteration bound for infeasible interior-point methods.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Miao, J.: A quadratically convergent \(\mathcal{O}((1+\kappa)\sqrt{n}L)\)-iteration algorithm for the P ∗(κ)-matrix linear complementarity problem. Math. Program. 69, 355–368 (1995)

    MATH  Google Scholar 

  4. 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  MATH  MathSciNet  Google Scholar 

  5. Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems. Lecture Notes in Comput. Sci., vol. 538. Springer, Berlin (1991)

    Google Scholar 

  6. Gurtuna, F., Petra, C., Potra, F., Shevchenko, O., Vancea, A.: Corrector-predictor methods for sufficient linear complementarity problems. Comput. Optim. Appl. 48, 453–485 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Wang, G.Q., Bai, Y.Q.: Polynomial interior-point algorithm for P ∗(κ) horizontal linear complementarity problem. J. Comput. Appl. Math. 233(2), 248–263 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  9. 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)

    Article  MATH  MathSciNet  Google Scholar 

  10. Asadi, S., Mansouri, H.: Polynomial interior-point algorithm for P ∗(κ)-horizontal linear complementarity problems. Numer. Algorithms 63(2), 385–398 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kheirfam, B.: A predictor-corrector interior-point algorithm for P ∗(κ) horizontal linear complementarity problem. Numer. Algorithms (2013). doi:10.1007/s11075-013-9738-3

    Google Scholar 

  12. Lustig, I.J.: Feasible issues in a primal-dual interior-point method. Math. Program. 67, 145–162 (1990)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Zhang, Y.: On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem. SIAM J. Optim. 4(1), 208–227 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mizuno, S.: Polynomiality of infeasible-interior-point algorithms for linear programming. Math. Program. 67(1), 109–119 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  16. Potra, F.A., Sheng, R.Q.: A large-step infeasible-interior-point method for the P ∗-matrix LCP. SIAM J. Optim. 7(2), 318–335 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  18. Roos, C.: A full-Newton step O(n) infeasible interior-point algorithm for linear optimization. SIAM J. Optim. 16(4), 1110–1136 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Mansouri, H., Zangiabadi, M., Pirhaji, M.: A full-Newton step O(n) infeasible-interior-point algorithm for linear complementarity problems. Nonlinear Anal., Real World Appl. 12(1), 545–561 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kheirfam, B.: A full NT-step infeasible interior-point algorithm for semidefinite optimization based on a self-regular proximity. ANZIAM J. 53, 48–67 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kheirfam, B.: Simplified infeasible interior-point algorithm for SDO using full Nesterov–Todd step. Numer. Algorithms 59(4), 589–606 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kheirfam, B.: A full-Newton step infeasible interior-point algorithm for linear complementarity problems based on a kernel function. Algorithmic Oper. Res. 7, 103–110 (2013)

    MathSciNet  Google Scholar 

  23. Kheirfam, B., Mahdavi-Amiri, N.: A full Nesterov–Todd step infeasible interior-point algorithm for symmetric cone linear complementarity problem. Bull. Iran. Math. Soc. (2013) in press

  24. Zhang, L., Sun, L., Xu, Y.: Simplified analysis for full-Newton step infeasible interior-point algorithm for semidefinite programming. Optimization 62(2), 169–191 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  25. Anitescu, M., Lesaja, G., Potra, F.A.: An infeasible-interior-point predictor–corrector algorithm for the P ∗-Geometric LCP. Appl. Math. Optim. 36(2), 203–228 (1997)

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  27. Gu, G., Mansouri, H., Zangiabadi, M., Bai, Y.Q., Roos, C.: Improved full-Newton step O(nL) infeasible interior-point method for linear optimization. J. Optim. Theory Appl. 145, 271–288 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  28. Potra, F., Stoer, J.: On a class of superlinearly convergent polynomial time interior point methods for sufficient LCP. SIAM J. Optim. 20(3), 1333–1363 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kheirfam, B.: A full-Newton step infeasible interior-point algorithm for P ∗(κ)-horizontal linear complementarity problems. Submitted

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Authors and Affiliations

  1. Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

    Behrouz Kheirfam

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  1. Behrouz Kheirfam
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Correspondence to Behrouz Kheirfam.

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Communicated by Florian A. Potra.

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Kheirfam, B. A New Complexity Analysis for Full-Newton Step Infeasible Interior-Point Algorithm for Horizontal Linear Complementarity Problems. J Optim Theory Appl 161, 853–869 (2014). https://doi.org/10.1007/s10957-013-0457-7

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  • DOI: https://doi.org/10.1007/s10957-013-0457-7

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