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Globally Convergent Broyden-Like Methods for Semismooth Equations and Applications to VIP, NCP and MCP

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Abstract

In this paper, we propose a general smoothing Broyden-like quasi-Newton method for solving a class of nonsmooth equations. Under appropriate conditions, the proposed method converges to a solution of the equation globally and superlinearly. In particular, the proposed method provides the possibility of developing a quasi-Newton method that enjoys superlinear convergence even if strict complementarity fails to hold. We pay particular attention to semismooth equations arising from nonlinear complementarity problems, mixed complementarity problems and variational inequality problems. We show that under certain conditions, the related methods based on the perturbed Fischer–Burmeister function, Chen–Harker–Kanzow–Smale smoothing function and the Gabriel–Moré class of smoothing functions converge globally and superlinearly.

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References

  1. X. Chen, Superlinear convergence of smoothing quasi-Newton methods for nonsmooth equations, J. Comput. Appl. Math. 80 (1997) 105–126.

    Google Scholar 

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

    Google Scholar 

  3. C. Chen and O.L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl. 5 (1996) 97–138.

    Google Scholar 

  4. X. Chen and L. Qi, A parameterized Newton method and a quasi-Newton method for nonsmooth equations, Comput. Optim. Appl. 3 (1994) 157–179.

    Google Scholar 

  5. 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. Comput. 67 (1998) 519–540.

    Google Scholar 

  6. F.H. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, NY, 1983).

    Google Scholar 

  7. T. De Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Programming 75 (1996) 407–440.

    Google Scholar 

  8. J.E. Dennis, Jr. and J.J. Moré, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comput. 28 (1974) 549–560.

    Google Scholar 

  9. J.E. Dennis, Jr. and J.J. Moré, Quasi-Newton methods: Motivation and theory, SIAM Rev. 19 (1977) 46–89.

    Google Scholar 

  10. F. Facchinei and J. Soares, A new merit function for nonlinear complementarity problems and related algorithm, SIAM J. Optim. 7 (1997) 225–247.

    Google Scholar 

  11. A. Fischer, A special Newton-type optimization method, Optimization 24 (1992) 269–284.

    Google Scholar 

  12. A. Fischer, A Newton-type method for positive semidefinite linear complementarity problems, J. Optim. Theory Appl. 86 (1995) 585–608.

    Google Scholar 

  13. R. Fletcher, Practical Methods of Optimization, 2nd edn. (Wiley, Chichester, 1987).

    Google Scholar 

  14. M. Fukushima and L. Qi, eds., Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods (Kluwer Academic, Boston, MA, 1998).

    Google Scholar 

  15. S.A. Gabriel and J.J. Moré, Smoothing of mixed complementarity problems, in: Complementarity and Variational Problems: State of the Art, eds. M.C. Ferris and J.S. Pang (SIAM, Philadelpha, PA, 1997) pp. 105–116.

    Google Scholar 

  16. C. Geiger and C. Kanzow, On the solution of monotone complementarity problems, Comput. Optim. Appl. 5 (1996) 155–173.

    Google Scholar 

  17. C.M. Ip and J. Kyparisis, Local convergence of quasi-Newton methods for B-differentiable equations, Math. Programming 56 (1992) 71–89.

    Google Scholar 

  18. H. Jiang and L. Qi, A new nonsmooth equations approach to nonlinear complementarity problems, SIAM J. Control Optim. 35 (1997) 178–193.

    Google Scholar 

  19. C. Kanzow and M. Fukushima, Equivalence of the generalized complementarity problem to differentiable unconstrained minimization, J. Optim. Theory Appl. 90 (1996) 581–603.

    Google Scholar 

  20. C. Kanzow and H. Jiang, A continuation method for (strongly) monotone variational inequalities, Math. Programming 81 (1998) 103–125.

    Google Scholar 

  21. C. Kanzow and H. Pieper, Jacobian smoothing methods for general nonlinear complementarity problems, SIAM J. Optim. 9 (1999) 342–373.

    Google Scholar 

  22. M. Kojima and S. Shindo, Extension of Newton and quasi-Newton methods to systems of PC1 equations, J. Oper. Res. Soc. Japan 29 (1986) 352–374.

    Google Scholar 

  23. D. Li, Global convergence of nonsingular Broyden's method for solving unconstrained optimization, Chinese J. Numer. Math. Appl. 17 (4) (1995) 85–94.

    Google Scholar 

  24. D. Li and M. Fukushima, A derivative-free line search and global convergence of Broyden-like method for nonlinear equations, Optim. Methods Softw. 13 (2000) 181–201.

    Google Scholar 

  25. D. Li and M. Fukushima, Smoothing Newton and quasi-Newton methods for mixed complementarity problems, Comput. Optim. Appl. 17 (2000) 203–230.

    Google Scholar 

  26. D. Li and M. Fukushima, Broyden-like methods for variational inequality and nonlinear complementarity problems with global and superlinear convergence, Technical Report 98016, Department of Applied Mathematics and Physics, Kyoto University (September 1998).

  27. R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim. 15 (1977) 957–972.

    Google Scholar 

  28. F.A. Potra, L. Qi and D. Sun, Secant methods for semismooth equations, Numer. Math. 80 (1998) 305–324.

    Google Scholar 

  29. L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res. 18 (1993) 227–244.

    Google Scholar 

  30. L. Qi and X. Chen, A globally convergent successive approximation method for severely nonsmooth equations, SIAM J. Control Optim. 33 (1995) 402–418.

    Google Scholar 

  31. L. Qi, and D. Sun, A survey of some nonsmooth equations and smoothing Newton methods, in: Progress in Optimization, eds. A. Eberhard, R. Hill, D. Ralph and B.M. Glover (Kluwer Academic, Dordrecht, 1999) pp. 121–146.

    Google Scholar 

  32. 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 87 (2000) 1–35.

    Google Scholar 

  33. D. Sun and J. Han, Newton and quasi-Newton methods for a class of nonsmooth equations and related problems, SIAM J. Optim. 7 (1997) 463–480.

    Google Scholar 

  34. N. Xiu and J. Zhang, A smoothing Gauss-Newton method for the generalized HLCP, J. Comput. Appl. Math. 129 (2001) 195–208.

    Google Scholar 

  35. N. Yamashita and M. Fukushima, Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems, Math. Programming 76 (1997) 469–491.

    Google Scholar 

  36. S. Zhou, D. Li and J. Zeng, A successive approximation quasi-Newton process for nonlinear complementarity problems, in: Recent Advances in Nonsmooth Optimization, eds. D.Z. Du, L. Qi and R.S. Womersley (World Scientific, Singapore, 1995) pp. 459–472.

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Hunan University, Changsha, 410082, China

    Dong-Hui Li

  2. Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan

    Masao Fukushima

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  1. Dong-Hui Li
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  2. Masao Fukushima
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Li, DH., Fukushima, M. Globally Convergent Broyden-Like Methods for Semismooth Equations and Applications to VIP, NCP and MCP. Annals of Operations Research 103, 71–97 (2001). https://doi.org/10.1023/A:1012996232707

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