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A step-truncated method in a wide neighborhood interior-point algorithm for linear programming

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Abstract

In path-following interior-point algorithms, the predictor–corrector algorithms have quadratic convergence for the nondegenerate case, such that the obtained solution is of high precision. However, in each iteration it needs two steps: predictor and corrector, so the actual number of iterations increases. In this paper, we propose a step-truncated technique in a path-following interior-point algorithm for linear programming, based on Ai-Zhang’s wide neighborhood and Mehrotra’s second-order corrector. More specifically, if the step size is larger than one at the current iterate, we truncate this step size by setting it to one; and then start a new procedure as follows: reduce the centering parameter until the new iterate reaches the boundary of the wide neighborhood. By the proposed step-truncated technique, the new algorithm obtains the quadratical convergence as same as the predictor–corrector algorithms for the nondegenerate case, while the corrector step is not required. Numerical results show the advantages of the proposed step-truncated method under the requirement of high precision.

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References

  1. Ai, W.: Neighborhood-following algorithms for linear programming. Sci. China Ser. A Math. 47, 812–820 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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, 400–417 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Bai, Y.Q., Lesaja, G., Roos, C., Wang, G.Q., Ghami, M.E.: A class of large-update and small-update primal-dual interior-point algorithms for linear optimization. J. Optim. Theory Appl. 138, 341–359 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Darvay, Z., Kheirfam, B., Rigó, P.: A new wide neighborhood primal-dual second-order corrector algorithm for linear optimization. Optim. Lett. 14, 1747–1763 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. Darvay, Z., Rigó, P.: Large-step interior-point algorithm for linear optimization based on a new wide neighbourhood. CEJOR 26, 551–563 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Gonzaga, C.C.: Path-following methods for linear programming. SIAM Rev. 34, 167–224 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gonzaga, C.C.: Complexity of predictor-corrector algorithms for LCP based on a large neighborhood of the central path. SIAM J. Optim. 10, 183–194 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gonzaga, C.C., Tapia, R.A.: On the quadratic convergence of the simplified Mizuno-Todd-Ye algorithm for linear programming. SIAM J. Optim. 7, 66–85 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Güler, O., Ye, Y.: Convergence behavior of interior-point algorithms. Math. Program. 60, 215–228 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hung, P.F., Ye, Y.: An asymptotical O(\(\sqrt{n}\)L)-iteration path-following linear programming algorithm that uses wide neighborhoods. SIAM J. Optim. 6, 570–586 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, C., Liu, H., Cong, W.: An O(\(\sqrt{n}\)L) iteration primal-dual second-order corrector algorithm for linear programming. Optim. Lett. 5, 729–743 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, H., Liu, X., Liu, C.: Mehrotra-type predictor-corrector algorithms for sufficient linear complementarity problem. Appl. Numer. Math. 62, 1685–1700 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, H., Yang, X., Liu, C.: A new wide neighborhood primal-dual infeasible-interior-point method for symmetric cone programming. J. Optim. Theory Appl. 158, 796–815 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Luo, Z., Wu, S.: A modified predictor-corrector method for linear programming. Comput. Optim. Appl. 3, 83–91 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ma, X., Liu, H.: A superlinearly convergent wide-neighborhood predictor-corrector interior-point algorithm for linear programming. J. Appl. Math. Comput. 55, 669–682 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Peng, J., Roos, C., Terlaky, T.: Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms. Princeton University Press, Princeton (2002)

    MATH  Google Scholar 

  23. Potra, F.A.: Interior point methods for sufficient horizontal LCP in a wide neighborhood of the central path with best known iteration complexity. SIAM J. Optim. 24, 1–28 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Salahi, M., Peng, J., Terlaky, T.: On Mehrotra-type predictor–corrector algorithms. SIAM J. Optim. 18, 1377–1397 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Salahi, M., Mahdavi-Amiri, N.: Polynomial time second order Mehrotra-type predictor-corrector algorithms. Appl. Math. Comput. 183, 646–658 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Wright, S.J.: Primal-Dual Interior-Point Methods, vol. 54. SIAM, New York (1997)

    Book  MATH  Google Scholar 

  27. Ye, Y., Anstreicher, K.: On quadratic and \({O}(\sqrt{n}{L})\) convergence of a predictor-corrector algorithm for LCP. Math. Program. 62, 537–551 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ye, Y.: Interior Point Algorithms: Theory and Analysis. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  29. Ye, Y., Güler, O., Tapia, R.A., Zhang, Y.: A quadratically convergent O(\(\sqrt{n}\)L)-iteration algorithm for linear programming. Math. Program. 59, 151–162 (1993)

    Article  MATH  Google Scholar 

  30. Ye, Y., Todd, M.J., 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 

  31. Zhang, Y., Tapia, R.A.: Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited. J. Optim. Theory Appl. 73, 229–242 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, Y., Tapia, R.A.: A superlinearly convergent polynomial primal-dual interior-point algorithm for linear programming. SIAM J. Optim. 3, 118–133 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, Y., Tapia, R.A., Dennis, J.E.: On the superlinear and quadratic convergence of primal-dual interior point linear programming algorithms. SIAM J. Optim. 2, 304–324 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grants Nos. 11871115, 11971073, 11771056). This work was also supported by Beijing Natural Science Foundation (Grants No. Z220004).

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  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Jianbin Wang, Jianhua Yuan & Wenbao Ai

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  1. Jianbin Wang
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Correspondence to Wenbao Ai.

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Wang, J., Yuan, J. & Ai, W. A step-truncated method in a wide neighborhood interior-point algorithm for linear programming. Optim Lett 17, 1455–1468 (2023). https://doi.org/10.1007/s11590-022-01941-2

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