# Strong convergence of extragradient method for generalized variational inequalities in Hilbert space

- Haibin Chen
^{1}Email author, - Yiju Wang
^{}and - Gang Wang
^{}

**2014**:223

https://doi.org/10.1186/1029-242X-2014-223

© Chen et al.; licensee Springer. 2014

**Received: **15 December 2013

**Accepted: **23 May 2014

**Published: **3 June 2014

## Abstract

In this paper, we present a new type of extra-gradient method for generalized variational inequalities with multi-valued mapping in an infinite-dimensional Hilbert space. For this method, the generated sequence possesses an expansion property with respect to the initial point, and the existence of the solution to the problem can be verified through the behavior of the generated sequence. Furthermore, under mild conditions, we show that the generated sequence of the method strongly converges to the solution of the problem which is closest to the initial point.

**MSC:**90C30, 15A06.

## Keywords

## 1 Introduction

*F*be a multi-valued mapping from ℋ into ${2}^{\mathcal{H}}$ with nonempty values, where ℋ is a real Hilbert space. Let

*X*be a nonempty, closed and convex subset of the Hilbert space ℋ. The generalized variational inequality problem, abbreviated as GVIP, is to find a vector ${x}^{\ast}\in X$ such that there exists ${\omega}^{\ast}\in F({x}^{\ast})$ satisfying

where $\u3008\cdot ,\cdot \u3009$ stands for the inner product of vectors in ℋ. If the multi-valued mapping *F* is a single-valued mapping from ℋ to ℋ, then the GVIP collapses to the classical variational inequality problem [1, 2].

*etc.*and have received much attention in the past decades [3–11]. It is well known that the extra-gradient method [5, 12] is a popular solution method, which has a contraction property,

*i.e.*, the generated sequence ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$ by the method satisfies

*i.e.*,

Furthermore, we establish the strong convergence of the method in the case that the solution set ${X}^{\ast}$ is nonempty, and we show that the generated sequence ${\{\parallel {x}^{k}\parallel \}}_{k=0}^{\mathrm{\infty}}$ diverges to infinity if the solution set is empty.

The rest of this paper is organized as follows. In Section 2, we give some related concepts and conclusions needed in the subsequent analysis. In Section 3, we present our designed algorithm and establish the convergence of the algorithm.

## 2 Preliminaries

*K*be a nonempty, closed, and convex subset in ℋ. A point ${y}_{0}\in K$ is said to be the orthogonal projection of

*x*onto

*K*if it is the closest point to

*x*in

*K*,

*i.e.*,

and we denote ${y}_{0}$ by ${P}_{K}(x)$. The well-known properties of the projection operator are as follows.

**Lemma 2.1** [15]

*Let*

*K*

*be a nonempty*,

*closed*,

*and convex subset in*ℋ.

*Then for any*$x,y\in \mathcal{H}$,

*and*$z\in K$,

*the following statements hold*:

- (i)
$\u3008{P}_{K}(x)-x,z-{P}_{K}(x)\u3009\ge 0$;

- (ii)
${\parallel {P}_{K}(x)-{P}_{K}(y)\parallel}^{2}\le {\parallel x-y\parallel}^{2}-{\parallel {P}_{K}(x)-x+y-{P}_{K}(y)\parallel}^{2}$.

**Remark 2.1**In fact, (i) in Lemma 2.1 also provides a sufficient condition for a vector

*u*to be the projection of the vector

*x*,

*i.e.*, $u={P}_{K}(x)$ if and only if

**Definition 2.1**Let

*K*be a nonempty subset of ℋ. The multi-valued mapping $F:K\to {2}^{\mathcal{H}}$ is said to be

- (i)monotone if and only if$\u3008u-v,x-y\u3009\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x,y\in K,u\in F(x),v\in F(y);$
- (ii)pseudo-monotone if and only if, for any $x,y\in K$, $u\in F(x)$, $v\in F(y)$,$\u3008u,y-x\u3009\ge 0\phantom{\rule{1em}{0ex}}\u27f9\phantom{\rule{1em}{0ex}}\u3008v,y-x\u3009\ge 0.$

To proceed, we present the definition of maximal monotone multi-valued mapping *F*.

**Definition 2.2**Let

*K*be a nonempty subset of ℋ. The multi-valued mapping $F:K\to {2}^{\mathcal{H}}$ is said to be a maximal monotone operator if

*F*is monotone and the graph

is not properly contained in the graph of any other monotone operator.

It is clear that a monotone multi-valued mapping *F* is maximal if and only if, for any $(x,u)\in K\times \mathcal{H}$ such that $\u3008u-v,x-y\u3009\ge 0$, $\mathrm{\forall}(y,v)\in G(F)$, then $u\in F(x)$.

**Definition 2.3**Let

*K*be a nonempty, closed, and convex subset of the Hilbert space ℋ. A multi-valued mapping $F:K\to {2}^{\mathcal{H}}$ is said to be

- (i)
upper semi-continuous at $x\in K$ if for every open set

*V*containing $F(x)$, there is an open set*U*containing*x*such that $F(y)\subset V$ for all $y\in K\cap U$; - (ii)
lower semi-continuous at $x\in K$ if given any sequence ${x}^{k}$ converging to

*x*and any $y\in F(x)$, there exists a sequence ${y}^{k}\in F({x}^{k})$ that converges to*y*; - (iii)
continuous at $x\in K$ if it is both upper semi-continuous and lower semi-continuous at

*x*.

Throughout this paper, we assume that the multi-valued mapping $F:X\to {2}^{\mathcal{H}}$ is maximal monotone and continuous on *X* with nonempty compact convex values, where $X\subseteq \mathcal{H}$ is a nonempty, closed, and convex set.

## 3 Main results

Then the projection residue $r(x,\xi )$ can verify the solution set of problem (1.1).

**Proposition 3.1**

*Point*$x\in X$

*solves problem*(1.1)

*if and only if*$r(x,\xi )=0$

*i*.

*e*.,

Now, we give the description of the designed algorithm for problem (1.1), whose basic idea is as follows. At each step of the algorithm, compute the projection residue $r({x}^{k},{\xi}^{k})$ at iterate ${x}^{k}$. If it is a zero vector for some ${\xi}^{k}\in F({x}^{k})$, then stop with ${x}^{k}$ being a solution of problem (1.1); otherwise, find a trial point ${y}^{k}$ by a back-tracking search at ${x}^{k}$ along the residue $r({x}^{k},{\xi}^{k})$, and the new iterate is obtained by projecting ${x}^{0}$ onto the intersection of *X* with two halfspaces, respectively, associated with ${y}^{k}$ and ${x}^{k}$. Repeat this process until the projection residue is a zero vector.

**Algorithm 3.1**

Step 0: Choose $\sigma ,\gamma \in (0,1)$, ${x}^{0}\in X$, $k=0$.

*m*satisfying $\mathrm{\exists}{\zeta}^{k}\in F({x}^{k}-{\gamma}^{m}r({x}^{k},{\xi}^{k}))$ such that

Set $k=k+1$ and go to Step 1.

The following conclusion addresses the feasibility of the stepsize rule (3.1), *i.e.*, the existence of point ${\zeta}^{k}$.

**Lemma 3.1** *If* ${x}^{k}$ *is not a solution of problem* (1.1), *then there exists a smallest non*-*negative integer* *m* *satisfying* (3.1).

*Proof*By the definition of $r({x}^{k},{\xi}^{k})$ and Lemma 2.1, it follows that

*F*is continuous, we know that there exists ${\zeta}^{m}\in F({x}^{k}-{\gamma}^{m}r({x}^{k},{\xi}^{k}))$ such that

This completes the proof. □

**Lemma 3.2**

*Suppose the solution set*${X}^{\ast}$

*is nonempty*,

*then the halfspace*${H}_{k}^{1}$

*in Algorithm*3.1

*separates the point*${x}^{k}$

*from the set*${X}^{\ast}$.

*Moreover*,

*Proof*By the definition of $r({x}^{k},{\xi}^{k})$ and Algorithm 3.1, we have

where ${\zeta}^{k}$ is a vector in $F({y}^{k})$. So, by the definition of ${H}_{k}^{1}$ and (3.3) it follows that ${x}^{k}\notin {H}_{k}^{1}$.

*F*is monotone on

*X*, one has

the desired result follows. □

Regarding the projection step, we shall prove that the set ${H}_{k}^{1}\cap {H}_{k}^{2}\cap X$ is always nonempty, even when the solution set ${X}^{\ast}$ is empty. Therefore the whole algorithm is well defined in the sense that it generates an infinite sequence ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$.

**Lemma 3.3** *If the solution set* ${X}^{\ast}\ne \mathrm{\varnothing}$, *then* ${X}^{\ast}\subseteq {H}_{k}^{1}\cap {H}_{k}^{2}\cap X$ *for all* $k\ge 0$.

*Proof*From the analysis in Lemma 3.2, it is sufficient to prove that ${X}^{\ast}\subseteq {H}_{k}^{2}$ for all $k\ge 0$. The proof will be given by induction. Obviously, if $k=0$,

Thus ${X}^{\ast}\subseteq {H}_{l+1}^{2}$. This shows that ${X}^{\ast}\subseteq {H}_{k}^{2}$ for all $k\ge 0$ and the desired result follows. □

**Lemma 3.4** *Suppose that* ${X}^{\ast}=\mathrm{\varnothing}$, *then* ${H}_{k}^{1}\cap {H}_{k}^{2}\cap X\ne \mathrm{\varnothing}$ *for all* $k\ge 0$.

*Proof*On the contrary, suppose ${k}_{0}>1$ is the smallest non-negative number such that

*M*such that

*i.e.*, there exists a point $\overline{x}\in int(X)\cap \{x\mid \parallel x-{x}^{0}\parallel <2M\}$ such that

which implies that the solution set ${X}^{\ast}$ is nonempty. We arrive at a contradiction and the desired result follows. □

In order to establish the convergence of the algorithm, we first show the expansion property of the algorithm w.r.t. the initial point.

**Lemma 3.5**

*Suppose Algorithm*3.1

*reaches an iteration*$k+1$.

*Then*

*Proof*By the iterative process of Algorithm 3.1, one has

*i.e.*,

and the proof is completed. □

From Lemma 3.4, Algorithm 3.1 generates an infinite sequence if the solution set of problem (1.1) is empty. More precisely, we have the following conclusion.

**Theorem 3.1**

*Suppose Algorithm*3.1

*generates an infinite sequence*${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$.

*Assume the sequence*${\{{\eta}_{k}\}}_{k=0}^{\mathrm{\infty}}$

*is bounded away from zero*.

*Then the generated sequence*${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$

*is bounded and its each weak accumulation point is a solution of problem*(1.1)

*if the solution set*${X}^{\ast}$

*is nonempty*.

*Otherwise*

*if the solution set* ${X}^{\ast}$ *is empty*.

*Proof*For the case that ${X}^{\ast}\ne \mathrm{\varnothing}$, by Lemma 3.3 and

for any ${x}^{\ast}\in {X}^{\ast}$. So, ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$ is a bounded sequence.

*F*is continuous with compact values, Proposition 3.11 in [17] implies that $\{F({y}^{k}):k\in N\}$ is a bounded set, and hence the sequence $\{{\zeta}^{k}:{\zeta}^{k}\in F({y}^{k})\}$ is bounded. Thus, by (3.5) and (3.7), it follows that

*i.e.*,

for some $\xi \in F(\overline{x})$ and $\overline{x}$ is a solution of problem (1.1).

Otherwise, a similar argument to the one above leads to the conclusion that any weak accumulation point of ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$ is a solution of problem (1.1), which contradicts the emptiness of the solution set, and the conclusion follows. □

We are in a position to prove strong convergence of Algorithm 3.1.

**Theorem 3.2** *Suppose Algorithm* 3.1 *generates an infinite sequence* ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$. *If the solution set* ${X}^{\ast}$ *is nonempty and the sequence* $\{{\eta}_{k}\}$ *is bounded away from zero*, *then the sequence* ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$ *converges strongly to a solution* ${x}^{\ast}$ *such that* ${x}^{\ast}={P}_{{X}^{\ast}}({x}^{0})$; *otherwise*, ${lim}_{k\to +\mathrm{\infty}}\parallel {x}^{k}-{x}^{0}\parallel =+\mathrm{\infty}$. *That is*, *the solution set of problem* (1.1) *is empty if and only if the sequence generated by Algorithm* 3.1 *diverges to infinity*.

*Proof*For the case that the solution set is nonempty, from Theorem 3.1, we know that the sequence ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$ is bounded and that every weak accumulate point ${x}^{\ast}$ of ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$ is a solution of problem (1.1). Let ${\{{x}^{{k}_{j}}\}}_{j=0}^{\mathrm{\infty}}$ be a weakly convergent subsequence of ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$, and let ${x}^{\ast}\in {X}^{\ast}$ be its weak limit. Let $\overline{x}={P}_{{X}^{\ast}}({x}^{0})$. Then by Lemma 3.3,

*j*. So, from the iterative procedure of Algorithm 3.1,

Since ${x}^{\ast}$ was taken as an arbitrary weak accumulation point of ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$, it follows that $\overline{x}$ is the unique weak accumulation point of this sequence. Since ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$ is bounded, the whole sequence ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$ weakly converges to $\overline{x}$. On the other hand, we have shown that every weakly convergent subsequence of ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$ converges strongly to $\overline{x}$. Hence, the whole sequence ${\{{x}^{k}\}}_{k=0}^{\mathrm{\infty}}$ converges strongly to $\overline{x}\in {X}^{\ast}$.

For the case that the solution set is empty, the conclusion can be obtained directly from Theorem 3.1. □

## Declarations

### Acknowledgements

This work was supported by the Natural Science Foundation of China (Grant Nos. 11171180, 11101303), and the Specialized Research Fund for the Doctoral Program of Chinese Higher Education (20113705110002). The authors would like to thank the reviewers for their careful reading, insightful comments, and constructive suggestions, which helped improve the presentation of the paper.

## Authors’ Affiliations

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.