Appl Math Optim 40:315–339 (1999)
1999 Springer-Verlag New York Inc.
A Regularization Newton Method for Solving
Nonlinear Complementarity Problems
School of Mathematics, University of New South Wales,
Sydney 2052, NSW, Australia
Communicated by J. Stoer
Abstract. In this paper we construct a regularization Newton method for solv-
ing the nonlinear complementarity problem (NCP(F)) and analyze its convergence
properties under the assumption that F is a P
-function. We prove that every ac-
cumulation point of the sequence of iterates is a solution of NCP(F) and that the
sequence of iterates is bounded if the solution set of NCP(F) is nonempty and
bounded. Moreover, if F is a monotone and Lipschitz continuous function, we prove
that the sequence of iterates is bounded if and only if the solution set of NCP(F)is
nonempty by setting t =
, where t ∈ [
, 1] is a parameter. If NCP(F) has a locally
unique solution and satisﬁes a nonsingularity condition, then the convergence rate is
superlinear (quadratic) without strict complementarity conditions. At each step, we
only solve a linear system of equations. Numerical results are provided and further
applications to other problems are discussed.
Key Words. Nonlinear complementarity problem, Nonsmooth equations, Regu-
larization, Generalized Newton method, Convergence.
AMS Classiﬁcation. 90C33, 90C30, 65H10.
This work was supported by the Australian Research Council.