Nonlinear Analysis: Hybrid Systems 4 (2010) 371–380
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Nonlinear Analysis: Hybrid Systems
journal homepage: www.elsevier.com/locate/nahs
An interior proximal point algorithm for nonlinear complementarity
problems
Abdellah Bnouhachem
a,b
, Muhammad Aslam Noor
c,∗
a
School of Management Science and Engineering, Nanjing University, Nanjing, 210093, PR China
b
Ibn Zohr University, ENSA, BP 32/S, Agadir, Morocco
c
Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan
article info
Article history:
Received 18 August 2009
Accepted 11 September 2009
Keywords:
Nonlinear complementarity problems
Pseudomonotone operators
Logarithmic–quadratic proximal methods
abstract
In this paper, we propose a new method for solving nonlinear complementarity problems
(NCP), where the underlying function F is pseudomonotone and continuous. The method
can be viewed as an extension of the method of Noor and Bnouhachem (2006) [13],
by performing an additional projection step at each iteration and another optimal step
length is employed to reach substantial progress in each iteration. We prove the global
convergence of the proposed method under some suitable conditions. Some numerical
results are given to illustrate the efficiency and the implementation of the new proposed
method.
© 2009 Elsevier Ltd. All rights reserved.
1. Introduction
The nonlinear complementarity problem (NCP) is to determine a vector x ∈ R
n
such that
x ≥ 0, F (x) ≥ 0 and x
T
F (x) = 0, (1.1)
where F is a nonlinear mapping from R
n
into itself. For the applications, formulation, numerical results and other aspects of
the complementarity problems, see [1–30] and the references therein.
Throughout this paper we assume that F is continuous and pseudomonotone with respect to R
n
+
and the solution set of
(1.1), denoted by Ω
∗
, is nonempty.
It is well known that NCP can be alternatively formulated as finding the zero point of the operator T (x) = F (x) + N
R
n
+
(x),
i.e., find x
∗
∈ R
n
+
such that 0 ∈ T (x
∗
), where N
R
n
+
(.) is the normal cone operator to R
n
+
defined by
N
R
n
+
(x) :=
{y : y
T
(v − x) ≤ 0, ∀v ∈ R
n
+
}, if x ∈ R
n
+
;
∅, otherwise.
One of the methods to find the zero of the operator T is the proximal point algorithms (PPA) introduced by Martinet [12],
which is as follows:
Algorithm 1.1. For a given x
0
∈ R
n
+
, the next point x
k+1
is solution of the following subproblem:
0 ∈ β
k
T (x) +∇
x
q(x, x
k
), (1.2)
where
q(x, x
k
) =
1
2
‖x − x
k
‖
2
(1.3)
∗
Corresponding author.
E-mail addresses: babedallah@yahoo.com (A. Bnouhachem), noormaslam@hotmail.com, noor@mscs.dal.ca (M.A. Noor).
1751-570X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.nahs.2009.09.010