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Quadratic Convergence of a Long-Step Interior-Point Method for Nonlinear Monotone Variational Inequality Problems

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Abstract

This paper offers an analysis on a standard long-step primal-dual interior-point method for nonlinear monotone variational inequality problems. The method has polynomial-time complexity and its q-order of convergence is two. The results are proved under mild assumptions. In particular, new conditions on the invariance of the rank and range space of certain matrices are employed, rather than restrictive assumptions like nondegeneracy.

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References

  1. Nesterov, Y., and Nemirovskii, A., Interior-Point Polynomial Algorithms in Convex Programming, SIAM Publications, Philadelphia, Pennsylvania, 1994.

    Google Scholar 

  2. Rockafellar, R. T., Lagrange Multipliers and Variational Inequalities, Variational Inequalities and Complementarity Problems: Theory and Applications, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, Wiley, New York, New York, pp. 303–322, 1980.

    Google Scholar 

  3. Tseng, P., Global Linear Convergence of a Path-Following Algorithm for Some Monotone Variational Inequality Problems, Journal of Optimization Theory and Applications, Vol. 75, pp. 265–279, 1992.

    Google Scholar 

  4. Ralph, D., and Wright, S., Superlinear Convergence of an Interior-Point Method for Monotone Variational Inequalities, Complementarity and Variational Problems: State of the Art, SIAM Publications, Philadelphia, Pennsylvania, pp. 345–385, 1997.

    Google Scholar 

  5. Sun, J., and Zhao, G., Global Linear and Local Quadratic Convergence of a Long-Step Adaptive-Mode Interior-Point Method for Some Monotone Variational Inequality Problems, SIAM Journal on Optimization, Vol. 8, pp. 123–139, 1998.

    Google Scholar 

  6. Zhang, Y., Tapia, R. A., and Dennis, J. E., On the Superlinear and Quadratic Convergence of Primal-Dual Interior-Point Linear Programming Algorithms, SIAM Journal on Optimization, Vol. 2, pp. 304–324, 1992.

    Google Scholar 

  7. Jansen, B., Interior-Point Technologies in Optimization: Complementarity, Sensitivity, and Algorithms, Report, Delft University of Technology, Delft, Netherlands, 1995.

    Google Scholar 

  8. Jansen, B., Roos, K., Terlaky, T., and Yoshise, A., Polynomiality of Primal-Dual Affine Scaling Algorithms for Nonlinear Complementarity Problems, Discussion Paper Series 648, Institute of Socio-Economic Planning, University of Tsukuba, Tsukuba, Japan, 1995.

    Google Scholar 

  9. Monteiro, R. D. C., and Zhou, F., On Superlinear Convergence of Infeasible-Interior-Point Algorithms for Linearly Constrained Convex Programs, Technical Report, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia, 1995.

    Google Scholar 

  10. Potra, F., and Ye, Y., Interior-Point Methods for Nonlinear Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 88, pp. 617–642, 1996.

    Google Scholar 

  11. Ye, Y., and Anstreicher, K., On Quadratic and O(√nL) Convergence of a Predictor-Corrector Algorithm for LCP, Mathematical Programming, Vol. 62, pp. 537–552, 1993.

    Google Scholar 

  12. Sun, and Zhu, J., A Predictor-Corrector Algorithm for Extended Linear-Quadratic Programming, Computer and Operations Research, Vol. 23, pp. 755–767, 1996.

    Google Scholar 

  13. Sun, J., Zhu, J., and Zhao, G., A Predictor-Corrector Algorithm for a Class of Nonlinear Saddle-Point Problems, SIAM Journal on Control and Optimization, Vol. 35 pp. 532–551, 1997.

    Google Scholar 

  14. Potra, F., On q-Order and r-Order of Convergence, Journal of Optimization Theory and Applications, Vol. 63, pp. 415–431, 1989.

    Google Scholar 

  15. Tseng, P., An Infeasible Path-Following Method for Monotone Complementarity Problems, Technical Report, Department of Mathematics, University of Washington, Seattle, Washington, 1994.

    Google Scholar 

  16. Wright, S., and Ralph, D., A Superlinear Infeasible-Interior-Point Algorithm for Monotone Nonlinear Complementarity Problems, Mathematics of Operations Research, Vol. 21, pp. 815–838, 1996.

    Google Scholar 

  17. Rockafellar, R. T., Network Flows and Monotropic Optimization, Wiley, New York, New York, 1984.

    Google Scholar 

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Authors and Affiliations

  1. Department of Decision Sciences, National University of Singapore, Kent Ridge, Republic of Singapore

    J. Sun (Senior Lecturer)

  2. Department of Mathematics, National University of Singapore, Kent Ridge, Republic of Singapore

    G. Y. Zhao (Lecturer)

Authors
  1. J. Sun
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  2. G. Y. Zhao
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Sun, J., Zhao, G.Y. Quadratic Convergence of a Long-Step Interior-Point Method for Nonlinear Monotone Variational Inequality Problems. Journal of Optimization Theory and Applications 97, 471–491 (1998). https://doi.org/10.1023/A:1022691020204

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