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Global and Local Superlinear Convergence Analysis of Newton-Type Methods for Semismooth Equations with Smooth Least Squares

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Book cover Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods

Part of the book series: Applied Optimization ((APOP,volume 22))

Abstract

The local superlinear convergence of the generalized Newton method for solving systems of nonsmooth equations has been proved by Qi and Sun under the semismooth condition and nonsingularity of the generalized Jacobian at the solution. Unlike the Newton method for systems of smooth equations, globalization of the generalized Newton method seems difficult to achieve in general. However, we show that global convergence analysis of various traditional Newton-type methods for systems of smooth equations can be extended to systems of nonsmooth equations with semismooth operators whose least squares objective is smooth. The value of these methods is demonstrated from their applications to various semismooth equation reformulations of nonlinear complementarity and related problems.

This work is supported by the Australian Research Council.

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References

  1. B. Chen, X. Chen and C. Kanzow, A penalized Fischer-Burmeister NCPfunction: Theoretical investigation and numerical results, Preprint 126, Institute of Applied Mathematics, University of Hamburg, Hamburg, Germany, 1997.

    Google Scholar 

  2. P.W. Christensen, A. Klarbring, J.S. Pang and N. Strömberg, Formulation and comparison of algorithms for frictional contact problems, manuscript, Department of Mechanical Engineering, Linköping University, S-581 83, Linköping (November 1996).

    Google Scholar 

  3. P.W. Christensen and J.S. Pang, Reformulation–Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi, eds., ( Kluwer Academic Publisher, Nowell, MA. USA, 1998 ), 81–116.

    Google Scholar 

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

    MATH  Google Scholar 

  5. T. De Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Mathematical Programming 75 (1996) 407–439.

    Article  MathSciNet  MATH  Google Scholar 

  6. J.E. Dennis and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equation, Prentice Hall, Englewood Cliffs, New Jersey, 1983.

    Google Scholar 

  7. S.P. Dirkse and M.C. Ferris, The PATH Solver: A non-monotone stabilization scheme for mixed complementarity problems, Optimization Methods and Software 5 (1995) 123–156.

    Article  Google Scholar 

  8. F. Facchinei, A. Fischer, C. Kanzow and J. Peng, A simply constrained optimization reformulation of KKT systems arising from variational inequalities, Applied Mathematics and Optimization,to appear.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Fischer, A Newton-type method for positive semidefinite linear complementarity problems, Journal of Optimization Theory and Applications 86 (1995) 585–608.

    MATH  Google Scholar 

  11. A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian functions, Mathematical Programming 76 (1997) 513–532.

    MathSciNet  MATH  Google Scholar 

  12. R. Fletcher, Practical Methods of Optimization John Wiley, 2nd Edition, 1987.

    Google Scholar 

  13. S.P. Han, J.-S. Pang and N. Rangaraj, Globally convergent Newton methods for nonsmooth equations, Mathematics of Operations Research 17 (1992) 586–607.

    Article  MathSciNet  MATH  Google Scholar 

  14. P.T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problem: A survey of theory, algorithms and applications, Mathematical Programming 48 (1990) 161–220.

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Jiang, Global convergence analysis of the generalized Newton and Gauss-Newton methods for the Fischer-Burmeister equation for the complementarity problem, Mathematics of Operations Research,to appear.

    Google Scholar 

  16. H. Jiang, Smoothed Fischer-Burmeister equation methods for the nonlinear complementarity problem, Manuscript, Department of Mathematics, The University of Melbourne, June 1997.

    Google Scholar 

  17. H. Jiang, M. Fukushima, L. Qi and D. Sun, A trust region method for solving generalized complementarity problems, SIAM Journal on Optimization 8 (1998) 140–157.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Jiang, L. Qi, X. Chen and D. Sun, Semismoothness and superlinear convergence in nonsmooth optimization and nonsmooth equations, in: G. Di Pillo and F. Giannessi editors, Nonlinear Optimization and Applications, Plenum Publishing Corporation, New York, 1996, pp. 197–212.

    Google Scholar 

  20. C. Kanzow, Some noninterior continuation methods for linear complementarity problems, SIAM Journal on Matrix Analysis and Applications 17 (1996) 851–868.

    Article  MathSciNet  MATH  Google Scholar 

  21. C. Kanzow, An inexact QP-based method for nonlinear complementarity problems, Preprint 120, Institute of Applied Mathematics, University of Hamburg, Hamburg, Germany, February, 1997.

    Google Scholar 

  22. C. Kanzow and H. Kleinmichel, A new class of semismooth Newton-type methods for nonlinear complementarity problems, Computational Optimization and Applications,to appear.

    Google Scholar 

  23. C. Kanzow, N. Yamashita and M. Fukushima, New NCP-functions and their properties, Journal of Optimization Theory and Applications 94 (1997) 115–135.

    Article  MathSciNet  MATH  Google Scholar 

  24. B. Kummer, Newton’s method for nondifferentiable functions, in: Guddat, J. et al., eds., Mathematical Research: Advances in Mathematical Optimization, Akademie’ Verlag, Berlin, 1988, 114–125.

    Google Scholar 

  25. B. Kummer, Newton’s Method Based on Generalized Derivatives for Non-smooth Functions: Convergence Analysis, In: W. Oettli and D Pallaschke, eds., Advances in Optimization, Springer-Verlag, Berlin, 1992, pp. 171–194.

    Google Scholar 

  26. Z.-Q. Luo and P. Tseng, A new class of merit functions for the nonlinear complementarity problem, in: M.C. Ferris and J.-S. Pang, eds., Complementarity and Variational Problems: State of the Art, SIAM Publications, 1996.

    Google Scholar 

  27. R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM Journal on Control and Optimization 15 (1977) 959–972.

    Article  MathSciNet  MATH  Google Scholar 

  28. J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.

    MATH  Google Scholar 

  29. J.-S. Pang, Newton’s methods for B-differentiable equations, Mathematics of Operations Research 15 (1990) 311–341.

    Article  MathSciNet  MATH  Google Scholar 

  30. J.-S. Pang, A B-differentiable equation based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity, and variational inequality problems, Mathematical Programming 51 (1991) 101131.

    Google Scholar 

  31. J.-S. Pang, Complementarity problems, in: R. Horst and P. Pardalos, eds., Handbook of Global Optimization, Kluwer Academic Publishers, Boston, 1994, pp. 271–338.

    Google Scholar 

  32. J.-S. Pang and L. Qi, Nonsmooth equations: Motivation and algorithms, SIAM Journal on Optimization 3 (1993) 443–465.

    Article  MathSciNet  MATH  Google Scholar 

  33. L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research 18 (1993) 227–244.

    Article  MathSciNet  MATH  Google Scholar 

  34. L. Qi, Regular almost smooth NCP and BVIP functions and globally and quadratical convergent generalized Newton methods for complementarity and variational inequality problems, Technical Report AMR 97/14, University of New South Wales, June 1997.

    Google Scholar 

  35. L. Qi and X. Chen, A globally convergent successive approximation methods for nonsmooth equations, SIAM Journal on Control and Optimization 33 (1995) 402–418.

    Article  MathSciNet  MATH  Google Scholar 

  36. L. Qi and H. Jiang, Semismooth Karush-Kuhn-Tucker equations and convergence analysis of Newton methods and Quasi-Newton methods for solving these equations, Mathematics of Operations Research 22 (1997) 301325.

    Google Scholar 

  37. L. Qi and J. Sun, A nonsmooth version of Newton’s method, Mathematical Programming 58 (1993) 353–368.

    Article  MathSciNet  MATH  Google Scholar 

  38. D. Ralph, Global convergence of damped Newton’s method for nonsmooth equations, via the path search, Mathematics of Operations Research 19 (1994) 352–389.

    Article  MathSciNet  MATH  Google Scholar 

  39. S.M. Robinson, Generalized equations, in: A. Bachem, M. Grötschel and B. Korte, eds., Mathematical Programming: The State of the Art, Springer-Verlag, Berlin, 1983, pp. 346–367.

    Chapter  Google Scholar 

  40. S.M. Robinson, Newton’s method for a class of nonsmooth equations, Set-valued Analysis 2 (1994) 291–305.

    Article  MathSciNet  MATH  Google Scholar 

  41. R.T. Rockafellar, Convex Analysis, Princeton, New Jersey, 1970.

    Google Scholar 

  42. D. Sun and R.S. Womersley, A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Gauss-Newton method, SIAM Journal on Optimization,to appear.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  44. N. Yamashita and M. Fukushima, Reformulation–Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi, eds., ( Kluwer Academic Publisher, Nowell, MA. USA, 1998 ), 405–420.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Melbourne, Parkville, Victoria, 3052, Australia

    Houyuan Jiang & Daniel Ralph

Authors
  1. Houyuan Jiang
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  2. Daniel Ralph
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Editor information

Editors and Affiliations

  1. Kyoto University, Kyoto, Japan

    Masao Fukushima

  2. The University of New South Wales, Sydney, Australia

    Liqun Qi

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Jiang, H., Ralph, D. (1998). Global and Local Superlinear Convergence Analysis of Newton-Type Methods for Semismooth Equations with Smooth Least Squares. In: Fukushima, M., Qi, L. (eds) Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods. Applied Optimization, vol 22. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-6388-1_10

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  • DOI: https://doi.org/10.1007/978-1-4757-6388-1_10

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4805-2

  • Online ISBN: 978-1-4757-6388-1

  • eBook Packages: Springer Book Archive

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