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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Global and superlinear convergence of the
smoothing Newton method and its application
to general box constrained variational inequalities


Authors: X. Chen, L. Qi and D. Sun
Journal: Math. Comp. 67 (1998), 519-540
MSC (1991): Primary 90C33, 90C30, 65H10
MathSciNet review: 1458218
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Abstract: The smoothing Newton method for solving a system of nonsmooth equations $F(x)=0$, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the $k$th step, the nonsmooth function $F$ is approximated by a smooth function $ f(\cdot, \varepsilon _k)$, and the derivative of $ f(\cdot, \varepsilon _k)$ at $x^k$ is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if $F$ is semismooth at the solution and $f$ satisfies a Jacobian consistency property. We show that most common smooth functions, such as the Gabriel-Moré function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is $P$-uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).


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Additional Information

X. Chen
Affiliation: School of Mathematics The University of New South Wales\ Sydney 2052, Australia
Email: X.Chen@unsw.edu.au

L. Qi
Affiliation: School of Mathematics The University of New South Wales\ Sydney 2052, Australia
Email: L.Qi@unsw.edu.au

D. Sun
Affiliation: School of Mathematics The University of New South Wales\ Sydney 2052, Australia
Email: sun@alpha.maths.unsw.edu.au

DOI: http://dx.doi.org/10.1090/S0025-5718-98-00932-6
PII: S 0025-5718(98)00932-6
Keywords: Variational inequalities, nonsmooth equations, smoothing approximation, smoothing Newton method, convergence
Received by editor(s): June 17, 1996
Received by editor(s) in revised form: January 9, 1997
Additional Notes: This work is supported by the Australian Research Council.
Article copyright: © Copyright 1998 American Mathematical Society