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TopIntroduction
The nonlinear complementarity problem (NCP) is to determine a vector such that
(1) where
is a nonlinear mapping. Throughout this paper we assume that
is continuous and monotone with respect to
and the solution set of (1) is nonempty. NCP has many important applications in engineering, economics, military operations planning, finance, medical treatment, supply chain management etc, (Auslender & Haddou, 1995; Das, 2009; Castagnoli & Favero, 2008; Ferris & Pang, 1997; Harker & Pang, 1990; Yan & Wang, 1997). Many numerical methods for solving NCP have been developed (Auslender & Haddou, 1995; Auslender et al., 1999; Bnouhachem & Noor, 2001; Burachik & Svaiter, 2001; Censor et al., 1994; Eckstein, 1998; Fischer, 1997; Guler, 1991; Iusem, 1998; Pang, 1995; Qi & Yang, 2002; Sun & Qi, 1999; Zhou, 2009).
NCP can be alternatively formulated as finding the zero point of an appropriate maximal monotone operator
(2) i.e., finding
such that
, where
is the normal cone operator to
defined by
(3)A well known method to find the zero point of a maximal monotone operator is the proximal point algorithm (PPA), which starts with any vector and and iteratively updates conforming the following problem:
(4)In order to obtain the new point, the subproblem (1.4) of PPA is equivalent to the following variational inequality problem:
Find such that
(5)