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

DOI:
https://doi.org/10.1090/S0025-5718-98-00932-6

MathSciNet review:
1458218

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Abstract | References | Similar Articles | Additional Information

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

MR Author ID:
196364

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

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