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Identification of the Optimal Active Set in a Noninterior Continuation Method for LCP

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Abstract

This paper concerns about the possibility of identifying the active set in a noninterior continuation method for solving the standard linear complementarity problem based on the algorithm and theory presented by Burke and Xu (J. Optim. Theory Appl. 112 (2002) 53). It is shown that under the assumptions of P-matrix and nondegeneracy, the algorithm requires at most Oρlog(β0μ0/τ)) iterations to find the optimal active set, where β0 is the width of the neighborhood which depends on the initial point, μ0> 0 is the initial smoothing parameter, ρ is a positive number which depends on the problem and the initial point, and τ is a small positive number which depends only on the problem.

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References

  1. Burke, J. and Xu, S. (1998), The global linear convergence of a non-interior path-following algorithm for linear complementarity problem, Math. Opera. Research 23, 719-734.

    Google Scholar 

  2. Burke, J. and Xu, S. (2000), A non-interior predictor-corrector path following algorithm for the monotone linear complementarity problem, Math. Programming 87, 113-130.

    Google Scholar 

  3. Burke, J. and Xu, S. (2002), Complexity of a noninterior path-following method for the linear complementarity problem, J. Optim. Theory Appl. 112, 53-76.

    Google Scholar 

  4. Chen, B. and Chen, X. (2000), A global linear and local quadratic continuation smoothing method for variational inequalities with box constraints, Compu. Optim. Appl. 17, 131-158.

    Google Scholar 

  5. Chen, B. and Chen, X. (1999), A global and local superlinear continuation-smoothing method for P0 + R0 and monotone NCP, SIAM J. Optimization 9, 624-645.

    Google Scholar 

  6. Chen, B. and Harker, P.T. (1993), A non-interior-point continuation method for linear complementarity problems, SIAM J. Matrix Anal. Appl. 14, 1168-1190.

    Google Scholar 

  7. Chen, B. and Harker, P.T. (1997), Smoothing approximations to nonlinear complementarity problems, SIAM J. Optim. 7, 403-420.

    Google Scholar 

  8. Chen, B. and Xiu, N. (1999), A global linear and local quadratic non-interior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing functions, SIAM J. Optimization, 9, 605-623.

    Google Scholar 

  9. Chen, B. and Xiu, N. (2001), Superlinear noninterior one-step continuation method for monotone LCP in the absence of strict complementarity, J. Optim. Theory Appl. 108, 317-332.

    Google Scholar 

  10. Chen, C.H. and Mangasarian, O.L. (1995), Smoothing methods for convex inequalities and linear complementarity problems, Math. Programming 71, 51-69.

    Google Scholar 

  11. Chen, C.H. and Mangasarian, O.L. (1996), A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optim. and Appl. 5, 97-138.

    Google Scholar 

  12. Chen, X., Qi, L. and Sun, D. (1998), Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Compu. 67, 519-540.

    Google Scholar 

  13. Chen, X. and Ye, Y. (1999), On homotopy-smoothing methods for variational inequalities, SIAM J. Control and Optimization, 37, 589-616.

    Google Scholar 

  14. Cottle, R.W., Pang, J.S. and Stone, R.E. (1992), The Linear Complementarity Problem, Academic Press, New York.

    Google Scholar 

  15. El-Bakry, A.S., Tapia, R.A. and Zhang, Y. (1994), A study of indicator for identifying zero variables in interior-point methods, SIAM Rev. 36, 45-72.

    Google Scholar 

  16. Facchinei, F., Fischer, A. and Kanzow, C. (1999), On the accurate identification of active constraints, SIAM J. Optimization 9, 14-32.

    Google Scholar 

  17. Facchinei, F., Fischer, A. and Kanzow, C. (2000), On the identification of zero variables in an interior-point framework, SIAM J. Optimization 10, 1058-1078.

    Google Scholar 

  18. Facchinei, F. and Kanzow, C. (1999), Beyond monotonicity in regularization methods for nonlinear complementarity problems, SIAM J. Control Optim., 37, 1150-1161.

    Google Scholar 

  19. Ferris, M.C. and Kanzow, C. (2000), Complementarity and related problems. In: Pardalos, P.M. and Resende, M.G.C. (eds.), Handbook on Applied Optimization, Oxford University Press, Oxford, UK.

    Google Scholar 

  20. Fukushima, M., Luo, Z.Q. and Pang, J.S. (1998), A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints, Compu. Optim. Appl. 10, 5-34.

    Google Scholar 

  21. Gabriel, S.A. and Moré, J.J. (1996), Smoothing of mixed complementarity problems. In: Ferris, M.C. and Pang, J.S. (eds.), Complementarity and Variational Problems: State of the Art, SIAM, Philadelphia, PA, pp. 105-116.

    Google Scholar 

  22. Gowda, M.S. and Tawhid, M.A. (1999), Existence and limiting behavior of trajectories associated with P0-equations, Compu. Optim. Appl. 12, 229-251.

    Google Scholar 

  23. Hotta, K. and Yoshise, A. (1999), Global convergence of a class of non-interior-point algorithm using Chen-Harker-Kanzow functions for nonlinear complementarity problems, Math. Programming, 86, 105-133.

    Google Scholar 

  24. Hotta, K., Inaba, M. and Yoshise, A. (2000), A complexity analysis of a smoothing method using CHKS-functions for monotone linear complementarity problems, Compu. Optim. Appl. 17, 183-201.

    Google Scholar 

  25. Illé, T., Peng, J., Roos, C. and Terlaky, T. (2000), A strongly polynomial rounding procedure yielding a maximally complementary solution for P *(κ) linear complementarity problem, SIAM J. Optimization 11, 320-340.

    Google Scholar 

  26. Kanzow, C. (1996), Some noninterior continuation methods for linear complementarity problems, SIAM J. Matrix Analysis and Applications, 17, 851-868.

    Google Scholar 

  27. Kanzow, C. and Pieper, H. (1999), Jacobian smoothing methods for general nonlinear complementarity problems, SIAM J. Optimization, 9, 342–372.

    Google Scholar 

  28. Mathias, R. (1989), An improved bound for a fundamental constant associated with a P-matrix, Appl. Math. Letters 2, 297-300.

    Google Scholar 

  29. Mathias, R. and Pang, J.S. (1990), Error bounds for the linear complementarity problem with a P-matrix, Linear Algebra Appl. 132, 123-136.

    Google Scholar 

  30. Peng, J.M. and Lin, Z. (1999), A noninterior continuation method for generalized linear complementarity problems, Math. Programming, 86, 533-563.

    Google Scholar 

  31. Qi, H.D. (2000), A regularized smoothing Newton method for box constrained variational inequality problems with P0 functions, SIAM J. Optimization, 10 315–330.

    Google Scholar 

  32. Qi, H.D. and Liao, L. (1999), A smoothing Newton method for extended vertical linear complementarity problems, SIAM J. Matrix Anal. Appl. 21, 45-66.

    Google Scholar 

  33. Qi, L. and Sun, D. (2000), Improving the convergence of non-interior point algorithms for nonlinear complementarity problems, Mathematics of Computation, 69, 283-304.

    Google Scholar 

  34. Qi, L. and Sun, D. Smoothing functions and a smoothing Newton method for complementarity and variational inequality problems, J. Optim. Theory Appl., to appear.

  35. Qi, L. and Sun, D. (1999), Nonsmooth equations and smoothing methods. In: Eberhard, A., Glover, B., Hill, R. and Ralph, D. (eds), Progress in Optimization: Contributions from Australasia, Kluwer Academic Publishers, Nowell, MA.

    Google Scholar 

  36. Qi, L., Sun, D. and Zhou, G. (2000), A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Programming 87, 1-35.

    Google Scholar 

  37. Ravindran, G. and Gowda, M.S. (2000), Regularization of P0-functions in box variational inequality problems, SIAM J. Optimization, 11, 748-760.

    Google Scholar 

  38. Sun, D. (1999), A regularization Newton method for solving nonlinear complementarity problems, Appl. Math. Optim., 40, 315-339.

    Google Scholar 

  39. Tseng, P. (1998), Analysis of a non-interior continuation method based on Chen-Mangasarian smoothing functions for complementarity problems. In: Fukushima, M. and Qi, L. (eds.), Reformulation-Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers, Nowell, MD, pp. 381-404.

    Google Scholar 

  40. Zhou, G., Sun, D. and Qi, L. (1998), Numerical experiments for a class of squared smoothing Newton methods for complementarity and variational inequality problems. In: Fukushima, M. and Qi, L. (eds.), Reformulation-Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers, Nowell, MD, pp. 421-441.

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Northern Jiaotong University, Beijing, 100044, China

    Naihua Xiu

  2. Department of Mathematics, City University of Hong Kong, Hong Kong, China

    Jianzhong Zhang

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  1. Naihua Xiu
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  2. Jianzhong Zhang
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Xiu, N., Zhang, J. Identification of the Optimal Active Set in a Noninterior Continuation Method for LCP. Journal of Global Optimization 26, 183–198 (2003). https://doi.org/10.1023/A:1023065422836

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