- Research Article
- Open access
- Published:
A New Method for Solving Monotone Generalized Variational Inequalities
Journal of Inequalities and Applications volume 2010, Article number: 657192 (2010)
Abstract
We suggest new dual algorithms and iterative methods for solving monotone generalized variational inequalities. Instead of working on the primal space, this method performs a dual step on the dual space by using the dual gap function. Under the suitable conditions, we prove the convergence of the proposed algorithms and estimate their complexity to reach an -solution. Some preliminary computational results are reported.
1. Introduction
Let be a convex subset of the real Euclidean space
,
be a continuous mapping from
into
, and
be a lower semicontinuous convex function from
into
. We say that a point
is a solution of the following generalized variational inequality if it satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ1_HTML.gif)
where denotes the standard dot product in
.
Associated with the problem (GVI), the dual form of this is expressed as following which is to find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ2_HTML.gif)
In recent years, this generalized variational inequalities become an attractive field for many researchers and have many important applications in electricity markets, transportations, economics, and nonlinear analysis (see [1–9]).
It is well known that the interior quadratic and dual technique are powerfull tools for analyzing and solving the optimization problems (see [10–16]). Recently these techniques have been used to develop proximal iterative algorithm for variational inequalities (see [17–22]).
In addition Nesterov [23] introduced a dual extrapolation method for solving variational inequalities. Instead of working on the primal space, this method performs a dual step on the dual space.
In this paper we extend results in [23] to the generalized variational inequality problem (GVI) in the dual space. In the first approach, a gap function is constructed such that
, for all
and
if and only if
solves (GVI). Namely, we first develop a convergent algorithm for (GVI) with
being monotone function satisfying a certain Lipschitz type condition on
. Next, in order to avoid the Lipschitz condition we will show how to find a regularization parameter at every iteration
such that the sequence
converges to a solution of (GVI).
The remaining part of the paper is organized as follows. In Section 2, we present two convergent algorithms for monotone and generalized variational inequality problems with Lipschitzian condition and without Lipschitzian condition. Section 3 deals with some preliminary results of the proposed methods.
2. Preliminaries
First, let us recall the well-known concepts of monotonicity that will be used in the sequel (see [24]).
Definition 2.1.
Let be a convex set in
, and
. The function
is said to be
(i)pseudomonotone on if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ3_HTML.gif)
(ii)monotone on if for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ4_HTML.gif)
(iii)strongly monotone on with constant
if for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ5_HTML.gif)
(iv)Lipschitz with constant on
(shortly
-Lipschitz), if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ6_HTML.gif)
Note that when is differentiable on some open set containing
, then, since
is lower semicontinuous proper convex, the generalized variational inequality (GVI) is equivalent to the following variational inequalities (see [25, 26]):
Find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ7_HTML.gif)
Throughout this paper, we assume that:
(A
1
) the interior set of , int 
is nonempty,
(A
2
) the set is bounded,
(A
3
) is upper semicontinuous on
, and
is proper, closed convex and subdifferentiable on
,
(A
4
) is monotone on
.
In special case , problem (GVI) can be written by the following.
Find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ8_HTML.gif)
It is well known that the problem (VI) can be formulated as finding the zero points of the operator , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ9_HTML.gif)
The dual gap function of problem (GVI) is defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ10_HTML.gif)
The following lemma gives two basic properties of the dual gap function (2.7) whose proof can be found, for instance, in [6].
Lemma 2.2.
The function is a gap function of (GVI), that is,
(i) for all
,
(ii) and
if and only if
is a solution to (DGVI). Moreover, if
is pseudomonotone then
is a solution to (DGVI) if and only if it is a solution to (GVI).
The problem may not be solvable and the dual gap function
may not be well-defined. Instead of using gap function
, we consider a truncated dual gap function
. Suppose that
fixed and
. The truncated dual gap function is defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ11_HTML.gif)
For the following consideration, we define as a closed ball in
centered at
and radius
, and
. The following lemma gives some properties for
.
Lemma 2.3.
Under assumptions (A1)–(A4), the following properties hold.
(i)The function is well-defined and convex on
.
(ii)If a point is a solution to (DGVI) then
.
(iii)If there exists such that
and
, and
is pseudomonotone, then
is a solution to (DGVI) (and also (GVI)).
Proof.
-
(i)
Note that
is upper semicontinuous on
for
and
is bounded. Therefore, the supremum exists which means that
is well-defined. Moreover, since
is convex on
and
is the supremum of a parametric family of convex functions (which depends on the parameter
), then
is convex on
-
(ii)
By definition, it is easy to see that
for all
. Let
be a solution of (DGVI) and
. Then we have
(2.9)
In particular, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ13_HTML.gif)
for all . Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ14_HTML.gif)
this implies .
-
(iii)
For some
,
means that
is a solution to (DGVI) restricted to
. Since
is pseudomonotone,
is also a solution to (GVI) restricted to
. Since
, for any
, we can choose
sufficiently small such that
(2.12)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ16_HTML.gif)
where (2.13) follows from the convexity of . Since
, dividing this inequality by
, we obtain that
is a solution to (GVI) on
. Since
is pseudomonotone,
is also a solution to (DGVI).
Let be a nonempty, closed convex set and
. Let us denote
the Euclidean distance from
to
and
the point attained this distance, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ17_HTML.gif)
As usual, is referred to the Euclidean projection onto the convex set
. It is well-known that
is a nonexpansive and co-coercive operator on
(see [27, 28]).
The following lemma gives a tool for the next discussion.
Lemma 2.4.
For any and for any
, the function
and the mapping
defined by (2.14) satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ19_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ20_HTML.gif)
Proof.
Inequality (2.15) is obvious from the property of the projection (see [27]). Now, we prove the inequality (2.16). For any
, applying (2.15) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ21_HTML.gif)
Using the definition of and noting that
and taking minimum with respect to
in (2.18), then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ22_HTML.gif)
which proves (2.16).
From the definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ23_HTML.gif)
Since , applying (2.15) with
instead of
and
for (2.20), we obtain the last inequality in Lemma 2.4.
For a given integer number , we consider a finite sequence of arbitrary points
, a finite sequence of arbitrary points
and a finite positive sequence
. Let us define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ24_HTML.gif)
Then upper bound of the dual gap function is estimated in the following lemma.
Lemma 2.5.
Suppose that Assumptions (A1)–(A4) are satisfied and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ25_HTML.gif)
Then, for any ,
(i), for all
,
.
(ii).
Proof.
-
(i)
We define
as the Lagrange function of the maximizing problem
. Using duality theory in convex optimization, then we have
(2.23)
 (ii) From the monotonicity of and (2.22), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ27_HTML.gif)
Combining (2.24), Lemma 2.5(i) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ28_HTML.gif)
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ29_HTML.gif)
3. Dual Algorithms
Now, we are going to build the dual interior proximal step for solving (GVI). The main idea is to construct a sequence such that the sequence
tends to 0 as
. By virtue of Lemma 2.5, we can check whether
is an
-solution to (GVI) or not.
The dual interior proximal step at the iteration
is generated by using the following scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ30_HTML.gif)
where and
are given parameters,
is the solution to (2.22).
The following lemma shows an important property of the sequence .
Lemma 3.1.
The sequence generated by scheme (3.1) satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ31_HTML.gif)
where ,
and
. As a consequence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ32_HTML.gif)
Proof.
We replace by
and
by
into (2.16) to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ33_HTML.gif)
Using the inequality (3.4) with ,
,
and noting that
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ34_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ35_HTML.gif)
From the subdifferentiability of the convex function to scheme (3.1), using the first-order necessary optimality condition, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ36_HTML.gif)
for all . This inequality implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ37_HTML.gif)
where .
We apply inequality (3.4) with ,
and
and using (3.8) to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ38_HTML.gif)
Combine this inequality and (3.6), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ39_HTML.gif)
On the other hand, if we denote , then it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ40_HTML.gif)
Combine (3.10) and (3.11), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ41_HTML.gif)
which proves (3.2).
On the other hand, from (3.9) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ42_HTML.gif)
Then the inequality (3.3) is deduced from this inequality and (3.6).
The dual algorithm is an iterative method which generates a sequence based on scheme (3.1). The algorithm is presented in detail as follows:
Algorithm 3.2.
One has the following.
Initialization:
Given a tolerance , fix an arbitrary point
and choose
,
. Take
and
.
Iterations:
For each , execute four steps below.
Step 1.
Compute a projection point by taking
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ43_HTML.gif)
Step 2.
Solve the strongly convex programming problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ44_HTML.gif)
to get the unique solution .
Step 3.
Find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ45_HTML.gif)
Set .
Step 4.
Compute
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ46_HTML.gif)
If , where
is a given tolerance, then stop.
Otherwise, increase by 1 and go back to Step 1.
Output:
Compute the final output as:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ47_HTML.gif)
Now, we prove the convergence of Algorithm 3.2 and estimate its complexity.
Theorem 3.3.
Suppose that assumptions (A1)–(A3) are satisfied and is
-Lipschitz continuous on
. Then, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ48_HTML.gif)
where is the final output defined by the sequence
in Algorithm 3.2. As a consequence, the sequence
converges to 0 and the number of iterations to reach an
-solution is
, where
denotes the largest integer such that
.
Proof.
From , where
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ49_HTML.gif)
Substituting (3.20) into (3.2), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ50_HTML.gif)
Using this inequality with for all
and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ51_HTML.gif)
If we choose for all
in (2.21), then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ52_HTML.gif)
Hence, from Lemma 2.5(ii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ53_HTML.gif)
Using inequality (3.22) and , it implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ54_HTML.gif)
Note that . It follows from the inequalities (3.24) and (3.25) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ55_HTML.gif)
which implies that . The termination criterion at Step 4,
, using inequality (2.26) we obtain
and the number of iterations to reach an
-solution is
.
If there is no the guarantee for the Lipschitz condition, but the sequences and
are uniformly bounded, we suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ56_HTML.gif)
then the algorithm can be modified to ensure that it still converges. The variant of Algorithm 3.2 is presented as Algorithm 3.4 below.
Algorithm 3.4.
One has the following.
Initialization:
Fix an arbitrary point and set
. Take
and
. Choose
for all
.
Iterations:
For each execute the following steps.
Step 1.
Compute the projection point by taking
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ57_HTML.gif)
Step 2.
Solve the strong convex programming problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ58_HTML.gif)
to get the unique solution .
Step 3.
Find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ59_HTML.gif)
Set .
Step 4.
Compute
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ60_HTML.gif)
If , where
is a given tolerance, then stop.
Otherwise, increase by 1, update
and go back to Step 1.
Output:
Compute the final output as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ61_HTML.gif)
The next theorem shows the convergence of Algorithm 3.4.
Theorem 3.5.
Let assumptions (A1)–(A3) be satisfied and the sequence be generated by Algorithm 3.4. Suppose that the sequences
and
are uniformly bounded by (3.27). Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ62_HTML.gif)
As a consequence, the sequence converges to 0 and the number of iterations to reach an
-solution is
.
Proof.
If we choose for all
in (2.21), then we have
. Since
, it follows from Step 3 of Algorithm 3.4 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ63_HTML.gif)
From (3.34) and Lemma 2.5(ii), for all we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ64_HTML.gif)
We define . Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ65_HTML.gif)
We consider, for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ66_HTML.gif)
Then derivative of is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ67_HTML.gif)
Thus is nonincreasing. Combining this with (3.36) and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ68_HTML.gif)
From Lemma 3.1, and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ69_HTML.gif)
Combining (3.39) and this inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ70_HTML.gif)
By induction on , it follows from (3.41) and
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ71_HTML.gif)
From (3.35) and (3.42), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ72_HTML.gif)
which implies that . The remainder of the theorem is trivially follows from (3.33).
4. Illustrative Example and Numerical Results
In this section, we illustrate the proposed algorithms on a class of generalized variational inequalities (GVI), where is a polyhedral convex set given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ73_HTML.gif)
where ,
. The cost function
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ74_HTML.gif)
where ,
is a symmetric positive semidefinite matrix and
. The function
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ75_HTML.gif)
Then is subdifferentiable, but it is not differentiable on
.
For this class of problem (GVI) we have the following results.
Lemma 4.1.
Let . Then
(i)if is
-strongly monotone on
, then
is monotone on
whenever
.
(ii)if is
-strongly monotone on
, then
is
-strongly monotone on
whenever
.
(iii)if is
-Lipschitz on
, then
is
-Lipschitz on
.
Proof.
Since is
-strongly monotone on
, that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ76_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ77_HTML.gif)
Then (i) and (ii) easily follow.
Using the Lipschitz condition, it is not difficult to obtain (iii).
To illustrate our algorithms, we consider the following data.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ78_HTML.gif)
with ,
,
. From Lemma 4.1, we have
is monotone on
. The subproblems in Algorithm 3.2 can be solved efficiently, for example, by using MATLAB Optimization Toolbox R2008a. We obtain the approximate solution
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ79_HTML.gif)
Now we use Algorithm 3.4 on the same variational inequalities except that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_Equ80_HTML.gif)
where the components of the
are defined by:
, with
randomly chosen in
and the
components of
are randomly chosen in
. The function
is given by Bnouhachem [19]. Under these assumptions, it can be proved that
is continuous and monotone on
.
With and the tolerance
, we obtained the computational results (see, the Table 1).
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F657192/MediaObjects/13660_2010_Article_2214_IEq313_HTML.gif)
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Acknowledgments
The authors would like to thank the referees for their useful comments, remarks and suggestions. This work was completed while the first author was staying at Kyungnam University for the NRF Postdoctoral Fellowship for Foreign Researchers. And the second author was supported by Kyungnam University Research Fund, 2010.
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Anh, P., Kim, J. A New Method for Solving Monotone Generalized Variational Inequalities. J Inequal Appl 2010, 657192 (2010). https://doi.org/10.1155/2010/657192
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DOI: https://doi.org/10.1155/2010/657192