# A New Method for Solving Monotone Generalized Variational Inequalities

- PhamNgoc Anh
^{1}and - JongKyu Kim
^{1}Email author

**2010**:657192

https://doi.org/10.1155/2010/657192

© Pham Ngoc Anh and Jong Kyu Kim. 2010

**Received: **11 May 2010

**Accepted: **4 October 2010

**Published: **12 October 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.

## Keywords

## 1. Introduction

where denotes the standard dot product in .

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

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]):

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
,

In special case , problem (GVI) can be written by the following.

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,

(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).

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 (A_{1})–(A_{4}), 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)).

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).

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.

Proof.

which proves (2.16).

Since , applying (2.15) with instead of and for (2.20), we obtain the last inequality in Lemma 2.4.

Then upper bound of the dual gap function is estimated in the following lemma.

Lemma 2.5.

- (i)

## 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.

where and are given parameters, is the solution to (2.22).

The following lemma shows an important property of the sequence .

Lemma 3.1.

Proof.

which proves (3.2).

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.

Step 2.

Step 3.

Step 4.

If , where is a given tolerance, then stop.

Otherwise, increase by 1 and go back to Step 1.

Output:

Now, we prove the convergence of Algorithm 3.2 and estimate its complexity.

Theorem 3.3.

_{1})–(A

_{3}) are satisfied and is -Lipschitz continuous on . Then, one has

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.

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 .

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.

Step 2.

Step 3.

Step 4.

If , where is a given tolerance, then stop.

Otherwise, increase by 1, update and go back to Step 1.

Output:

The next theorem shows the convergence of Algorithm 3.4.

Theorem 3.5.

_{1})–(A

_{3}) 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

As a consequence, the sequence converges to 0 and the number of iterations to reach an -solution is .

Proof.

which implies that . The remainder of the theorem is trivially follows from (3.33).

## 4. Illustrative Example and Numerical Results

Then is subdifferentiable, but it is not differentiable on .

For this class of problem (GVI) we have the following results.

Lemma 4.1.

(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.

Then (i) and (ii) easily follow.

Using the Lipschitz condition, it is not difficult to obtain (iii).

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 .

## Declarations

### 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.

## Authors’ Affiliations

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