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On the Rate of Local Convergence of High-Order-Infeasible-Path-Following Algorithms for P*-Linear Complementarity Problems

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Abstract

A simple and unified analysis is provided on the rate of local convergence for a class of high-order-infeasible-path-following algorithms for the P*-linear complementarity problem (P*-LCP). It is shown that the rate of local convergence of a ν-order algorithm with a centering step is ν + 1 if there is a strictly complementary solution and (ν + 1)/2 otherwise. For the ν-order algorithm without the centering step the corresponding rates are ν and ν/2, respectively. The algorithm without a centering step does not follow the fixed traditional central path. Instead, at each iteration, it follows a new analytic path connecting the current iterate with an optimal solution to generate the next iterate. An advantage of this algorithm is that it does not restrict iterates in a sequence of contracting neighborhoods of the central path.

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References

  1. J.F. Bonnans and C.C. Gonzaga, “Convergence of interior point algorithms for the linear complementarity problem,” Mathematics of Operations Research, vol. 21, pp. 1-25, 1996.

    Google Scholar 

  2. R.W. Cottle, J.-S. Pang, and R.E. Stone, The Linear Complementarity Problem, Academic Press: Boston, 1992.

    Google Scholar 

  3. M.S. Gowda, “On the extended linear complementarity problem,” Mathematical Programming, vol. 72, pp. 33-50, 1996.

    Google Scholar 

  4. P. Hung and Y. Ye, “An asymptotical O(\(\sqrt {nL} \))-iteration path-following linear programming algorithm that uses long steps,” SIAM Journal on Optimization, vol. 6, pp. 570-586, 1996.

    Google Scholar 

  5. M. Kojima, N. Megiddo, T. Noma, and A. Yoshise, A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Springer-Verlag: Berlin, 1996.

    Google Scholar 

  6. K. McShane, “Superlinearly convergent O(\(\sqrt {nL} \))-iteration interior point algorithms for LP and the monotone LCP,” SIAM Journal on Optimization, vol. 4, pp. 247-261, 1994.

    Google Scholar 

  7. S. Mizuno, “A superlinearly convergent infeasible-interior-point algorithm for geometrical LCPs without a strictly complementarity condition,” Mathematics of Operations Research, vol. 2, pp. 382-400, 1996.

    Google Scholar 

  8. S. Mizuno, F. Jarre, and J. Stoer, “A unified approach to infeasible-interior-point algorithm via geometrical linear complementarity problems,” J. Appl. Math. Opt., vol. 13, pp. 315-341, 1996.

    Google Scholar 

  9. R. Monteiro, I. Adler, and M. Resende, “A polynomial-time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power series extension,” Mathematics of Operations Research, vol. 15, pp. 191-214, 1990.

    Google Scholar 

  10. R. Monteiro and T. Tsuchiya, “Limiting behavior of the derivatives of certain trajectories associated with a monotone horizontal linear complementarity problem,” Mathematics of Operations Research, vol. 21, pp. 793-814, 1996.

    Google Scholar 

  11. F.A. Potra and R. Sheng, “A superlinearly convergent infeasible-interior-point algorithm for degenerate LCP,” Technical Report, Dept. of Math., The University of Iowa, USA, 1995.

    Google Scholar 

  12. J. Stoer and M. Wechs, “Infeasible-interior-point paths for sufficient linear complementarity problems and their analyticity,” Mathematical Programming, vol. 83, pp. 407-423, 1998.

    Google Scholar 

  13. J. Stoer, M. Wechs, and S. Mizuno, “High order infeasible-interior-point methods for solving sufficient linear complementarity problems,” Mathematics of Operations Research, vol. 23, pp. 832-862, 1998.

    Google Scholar 

  14. J.F. Sturm, “Superlinear convergence of an algorithm for monotone linear complementarity problems, when no strictly complementary solution exists,” Mathematics of Operations Research, vol. 24, pp. 72-94, 1999.

    Google Scholar 

  15. H. Väliaho, “P*-matrices are just sufficient,” Linear Algebra and Its Applications, vol. 239, pp. 103-108, 1996.

    Google Scholar 

  16. S.J. Wright and Y. Zhang, “A superquadratic infeasible-interior-point method for linear complementarity problems,” Mathematical Programming, vol. 73, pp. 269-289, 1996.

    Google Scholar 

  17. Y. Ye and K. Anstreicher, “On quadratic and O(\(\sqrt {nL} \)) convergence of a predictor-corrector algorithm for LCP,” Mathematical Programming, vol. 62, pp. 537-551, 1993.

    Google Scholar 

  18. G. Zhao, “Interior point algorithms for linear complementarity problems based on large neighborhoods of the central path,” SIAM Journal on Optimization, vol. 8, pp. 397-413, 1998.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, 119260, Singapore

    Gongyun Zhao & Jie Sun

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  1. Gongyun Zhao
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  2. Jie Sun
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Zhao, G., Sun, J. On the Rate of Local Convergence of High-Order-Infeasible-Path-Following Algorithms for P*-Linear Complementarity Problems. Computational Optimization and Applications 14, 293–307 (1999). https://doi.org/10.1023/A:1026492106091

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  • DOI: https://doi.org/10.1023/A:1026492106091

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