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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
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.
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, Linkö** University, S-581 83, Linkö** (November 1996).
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.
F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
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.
J.E. Dennis and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equation, Prentice Hall, Englewood Cliffs, New Jersey, 1983.
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.
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.
A. Fischer, A special Newton-type optimization method, Optimization 24 (1992) 269–284.
A. Fischer, A Newton-type method for positive semidefinite linear complementarity problems, Journal of Optimization Theory and Applications 86 (1995) 585–608.
A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian functions, Mathematical Programming 76 (1997) 513–532.
R. Fletcher, Practical Methods of Optimization John Wiley, 2nd Edition, 1987.
S.P. Han, J.-S. Pang and N. Rangaraj, Globally convergent Newton methods for nonsmooth equations, Mathematics of Operations Research 17 (1992) 586–607.
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.
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.
H. Jiang, Smoothed Fischer-Burmeister equation methods for the nonlinear complementarity problem, Manuscript, Department of Mathematics, The University of Melbourne, June 1997.
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.
H. Jiang and L. Qi, A new nonsmooth equations approach to nonlinear complementarity problems, SIAM Journal on Control and Optimization 35 (1997) 178–193.
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.
C. Kanzow, Some noninterior continuation methods for linear complementarity problems, SIAM Journal on Matrix Analysis and Applications 17 (1996) 851–868.
C. Kanzow, An inexact QP-based method for nonlinear complementarity problems, Preprint 120, Institute of Applied Mathematics, University of Hamburg, Hamburg, Germany, February, 1997.
C. Kanzow and H. Kleinmichel, A new class of semismooth Newton-type methods for nonlinear complementarity problems, Computational Optimization and Applications,to appear.
C. Kanzow, N. Yamashita and M. Fukushima, New NCP-functions and their properties, Journal of Optimization Theory and Applications 94 (1997) 115–135.
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.
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.
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.
R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM Journal on Control and Optimization 15 (1977) 959–972.
J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
J.-S. Pang, Newton’s methods for B-differentiable equations, Mathematics of Operations Research 15 (1990) 311–341.
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.
J.-S. Pang, Complementarity problems, in: R. Horst and P. Pardalos, eds., Handbook of Global Optimization, Kluwer Academic Publishers, Boston, 1994, pp. 271–338.
J.-S. Pang and L. Qi, Nonsmooth equations: Motivation and algorithms, SIAM Journal on Optimization 3 (1993) 443–465.
L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research 18 (1993) 227–244.
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.
L. Qi and X. Chen, A globally convergent successive approximation methods for nonsmooth equations, SIAM Journal on Control and Optimization 33 (1995) 402–418.
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.
L. Qi and J. Sun, A nonsmooth version of Newton’s method, Mathematical Programming 58 (1993) 353–368.
D. Ralph, Global convergence of damped Newton’s method for nonsmooth equations, via the path search, Mathematics of Operations Research 19 (1994) 352–389.
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.
S.M. Robinson, Newton’s method for a class of nonsmooth equations, Set-valued Analysis 2 (1994) 291–305.
R.T. Rockafellar, Convex Analysis, Princeton, New Jersey, 1970.
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.
N. Yamashita and M. Fukushima, Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems, Mathematical Programming 76 (1997) 469–491.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
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
Download citation
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