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A Global Linear and Local Quadratic Continuation Smoothing Method for Variational Inequalities with Box Constraints

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Abstract

In this paper, we propose a continuation method for box constrained variational inequality problems. The continuation method is based on the class of Gabriel-Moré smooth functions and has the following attractive features: It can start from any point; It has a simple and natural neighborhood definition; It solves only one approximate Newton equation at each iteration; It converges globally linearly and locally quadratically under nondegeneracy assumption at the solution point and other suitable assumptions. A hybrid method is also presented, which is shown to preserve the above convergence properties without the nondegeneracy assumption at the solution point. In particular, the hybrid method converges finitely for affine problems.

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References

  1. S.C. Billups, S.P. Dirkse, and M.C. Ferris, “A comparison of large-scale mixed complementarity problems solvers,” Comp. Optim. Appl., vol. 7, pp. 3-25, 1997.

    Google Scholar 

  2. J. Burke and S. Xu, “The global linear convergence of a non-interior path-following algorithm for linear complementarity problem,” Math. of Oper. Res., vol. 23, pp. 719-734, 1998.

    Google Scholar 

  3. B. Chen, “Finite convergence of nonsmooth equation based methods for affine variational inequalities,” Appl. Math. Lett., vol. 5, pp. 19-24, 1992.

    Google Scholar 

  4. B. Chen and P.T. Harker, “A non-interior-point continuation method for linear complementarity problems,” SIAM J. Matrix Anal. Appl., vol. 14, pp. 1168-1190, 1993.

    Google Scholar 

  5. B. Chen and P.T. Harker, “A non-interior-point continuation method for quadratic and linear programs,” SIAM J. Optim., vol. 3, pp. 503-515, 1993.

    Google Scholar 

  6. B. Chen and P.T. Harker, “A continuation method for monotone variational inequalities,” Math. Programming, vol. 69, pp. 237-253, 1995.

    Google Scholar 

  7. B. Chen and P.T. Harker, “Smooth approximations to nonlinear complementarity problems,” SIAM J. Optim., vol. 7, pp. 403-420, 1997.

    Google Scholar 

  8. B. Chen and N. Xiu, “A global linear and local quadratic non-interior continuation method for nonlinear complementarity problems based on Chen-Mangasarian smoothing function,” SIAM J. Optim., vol. 9, pp. 605-623, 1999.

    Google Scholar 

  9. C. Chen and O.L. Mangasarian, “Smoothing methods for convex inequalities and linear complementarity problems,” Math. Programming, vol. 71, pp. 51-69, 1995.

    Google Scholar 

  10. C. Chen and O.L. Mangasarian, “A class of smoothing functions for nonlinear and mixed complementarity problems,” Comp. Optim. and Appl., vol. 5, pp. 97-138, 1996.

    Google Scholar 

  11. X. Chen and L. Qi, “A parameterized Newton method and a Broyden-like method for solving nonsmooth equations,” Comp. Optim. Appl., vol. 3, pp. 157-179, 1994.

    Google Scholar 

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

    Google Scholar 

  13. X. Chen and Y. Ye, “On homotopy-smoothing methods for variational inequalities,” SIAM J. Con. and Optim., vol. 37, pp. 589-616, 1999.

    Google Scholar 

  14. F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley: New York, 1983.

    Google Scholar 

  15. B.C. Eaves, “On the basic theorem of complementarity,” Math. Programming, vol. 1, pp. 68-75, 1971.

    Google Scholar 

  16. M.C. Ferris and J.S. Pang, “Engineering and economic applications of complementarity problems,” SIAM Rev., vol. 39, pp. 669-713, 1997.

    Google Scholar 

  17. F. Facchinei, A. Fischer, and C. Kanzow, “A semismooth Newton method for variational inequalities: The case of box constraints,” in Complementarity and Variational Problems: State of the Art, M.C. Ferris and J.S. Pang (Eds.), SIAM: Philadelphia, Pennsylvania, 1996.

    Google Scholar 

  18. H. Jiang, “Global convergence analysis of the generalized newton and gauss-newton methods of the Fischer-Burmeister equation for the complementarity problem,” Math. of Oper. Res., vol. 24, pp. 529-543, 1999.

    Google Scholar 

  19. A. Fischer and C. Kanzow, “On the finite termination of an iterative method for linear complementarity problems,” Math. Programming, vol. 74, pp. 279-292, 1996.

    Google Scholar 

  20. R.M. Freund and S. Mizuno, “Interior point methods: current status and future directions,” OPTIMA Newsletter, vol. 51, pp. 1-9, 1996.

    Google Scholar 

  21. S.A. Gabriel, “A hybrid smoothing method for mixed nonlinear complementarity problems,” Comp. Optim. and Appl., vol. 9, pp. 153-173, 1998.

    Google Scholar 

  22. S.A. Gabriel and J.J. Moré, “Smoothing of mixed complementarity problems,” in Complementarity and Variational Problems: State of the Art, M.C. Ferris and J.S. Pang, (Eds.), SIAM: Philadelphia, Pennsylvania, 1997, pp. 105-116.

    Google Scholar 

  23. P.T. Harker and J.S. Pang, “Finite-dimensional variational inequality and nonlinear complementarity problem: A survey of theory, algorithms and applications,” Math. Programming, vol. 48, pp. 339-357, 1990.

    Google Scholar 

  24. K. Hotta and A. Yoshise, “Global convergence of a class of non-interior-point algorithms using Chen-Harker-Kanzow functions for nonlinear complementarity problems,” Math. Programming, vol. 86, pp. 105-133, 1999.

    Google Scholar 

  25. C. Kanzow, “Some noninterior continuation methods for linear complementarity problems,” SIAM J. Matrix Anal. Appl., vol. 17, pp. 851-868, 1996.

    Google Scholar 

  26. C. Kanzow and M. Fukushima, “Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities,” Math. Programming, vol. 83, pp. 55-87, 1998.

    Google Scholar 

  27. C. Kanzow and H. Jiang, “A continuation method for (strongly) monotone variational inequalities,” Math. Programming, vol. 81, pp. 140-157, 1998.

    Google Scholar 

  28. M. Kojima, N. Megiddo, and S. Mizuno, “A general framework of continuation methods for complementarity problems,” Math. of Oper. Res., vol. 18, pp. 945-963, 1993.

    Google Scholar 

  29. M. Kojima, N. Megiddo, and T. Noma, “Homotopy continuation methods for nonlinear complementarity problems,” Math. of Oper. Res., vol. 16, pp. 754-774, 1991.

    Google Scholar 

  30. J.S. Pang, “Newton's method for B-differentiable equations,” Math. Oper. Res., vol. 15, pp. 311-341, 1990.

    Google Scholar 

  31. J.S. Pang, “Complementarity problems,” in Handbook of Global Optimization, R. Horst and P. Pardalos (Eds.), Kluwer Academic Publishers: Boston, 1995, pp. 271-338.

    Google Scholar 

  32. L. Qi, “Convergence analysis of some algorithms for solving nonsmooth equations,” Math. Oper. Res., vol. 18, pp. 227-244, 1993.

    Google Scholar 

  33. L. Qi and X. Chen, “A globally convergent successive approximation method for severely nonsmooth equations,” SIAM J. Control Optim., vol. 33, pp. 402-418, 1995.

    Google Scholar 

  34. L. Qi and J. Sun, “A nonsmooth version of Newton's method,” Math. Programming, vol. 58, pp. 353-367, 1993.

    Google Scholar 

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

    Google Scholar 

  36. H. Sellami and S.M. Robinson, “Implementation of a continuation method for normal maps,” Math. Programming, vol. 76, pp. 563-578, 1997.

    Google Scholar 

  37. S. Smale, “Algorithms for solving equations,” in Proceedings of the International Congress of Mathematicians, Berkeley, California, 1986, pp. 172-195.

  38. D. Sun and R.S. Womersley, “A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Guass-Newton method,” SIAM J. Optim., vol. 9, pp. 388-413, 1999.

    Google Scholar 

  39. P. Tseng, “An infeasible path-following method for monotone complementarity problem,” SIAM J. Optim., vol. 7, pp. 386-402, 1997.

    Google Scholar 

  40. S. Wright and D. Ralph, “A superlinear infeasible-interior-point algorithm for monotone complementarity problems,” Math. Oper. Res., vol. 21, pp. 815-838, 1996.

    Google Scholar 

  41. S. Xu, “The global linear convergence of an infeasible non-interior path-following algorithm for complementarity problems with uniform P-functions,” Math. Programming, vol. 87, pp. 501-517, 2000.

    Google Scholar 

  42. S. Xu, “The global linear convergence and complexity of a non-interior path-following algorithm for monotone LCP based on Chen-Harker-Kanzow-Smale smooth functions,” Department of Mathematics, University of Washington, Seattle, WA 98195, Preprint, 1997.

    Google Scholar 

  43. Y. Ye and K. Anstreicher, “On quadratic and O(√NL) convergence of a predictor-corrector algorithm for LCP,” Math. Programming, vol. 62, pp. 537-552, 1993.

    Google Scholar 

  44. I. Zang, “A smoothing-out technique for min-max optimization,” Math. Programming, vol. 19, pp. 61-71, 1980.

    Google Scholar 

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Authors and Affiliations

  1. Department of Management and Decision Sciences, Washington State University, Pullman, WA, 99164-4736, USA

    Bintong Chen

  2. Department of Mathematics and Computer Science, Shimane University, Matsue, 690-8504, Japan

    Xiaojun Chen

Authors
  1. Bintong Chen
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  2. Xiaojun Chen
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Chen, B., Chen, X. A Global Linear and Local Quadratic Continuation Smoothing Method for Variational Inequalities with Box Constraints. Computational Optimization and Applications 17, 131–158 (2000). https://doi.org/10.1023/A:1026546230851

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