An Extension of Subgradient Method for Variational Inequality Problems in Hilbert Space

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Volume 2013, Article ID 531912,7pages http://dx.doi.org/10.1155/2013/531912

Research Article

An Extension of Subgradient Method for Variational Inequality

Problems in Hilbert Space

Xueyong Wang, Shengjie Li, and Xipeng Kou

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Correspondence should be addressed to Xueyong Wang; [email protected] Received 26 October 2012; Revised 30 January 2013; Accepted 1 February 2013 Academic Editor: Guanglu Zhou

Copyright © 2013 Xueyong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An extension of subgradient method for solving variational inequality problems is presented. A new iterative process, which relates to the fixed point of a nonexpansive mapping and the current iterative point, is generated. A weak convergence theorem is obtained for three sequences generated by the iterative process under some mild conditions.

1. Introduction

LetΩbe a nonempty closed convex subset of a real Hilbert

space𝐻, and let𝑓 : Ω → Ωbe a continuous mapping. The

variational inequality problem, denoted by VI(𝑓, Ω), is to find

a vector𝑥∗∈ Ω, such that

⟨𝑥 − 𝑥∗, 𝑓 (𝑥∗)⟩ ≥ 0, ∀ 𝑥 ∈ Ω. (1)

Throughout the paper, letΩ∗be the solution set of VI(𝑓, Ω),

which is assumed to be nonempty. In the special case when

Ω is the nonnegative orthant, (1) reduces to the nonlinear

complementarity problem. Find a vector𝑥∗∈ Ω, such that

𝑥∗≥ 0, 𝑓 (𝑥∗) ≥ 0, 𝑓(𝑥∗)𝑇𝑥∗ = 0. (2) The variational inequality problem plays an important role in optimization theory and variational analysis. There are numerous applications of variational inequalities in math-ematics as well as in equilibrium problems arising from

engineering, economics, and other areas in real life, see [1–16]

and the references therein. Many algorithms, which employ

the projection onto the feasible set Ω of the variational

inequality or onto some related sets in order to iteratively

reach a solution, have been proposed to solve (1). Korpelevich

[2] proposed an extragradient method for finding the saddle

point of some special cases of the equilibrium problem.

Solodov and Svaiter [3] extended the extragradient algorithm

through replying the setΩ by the intersection of two sets

related to VI(𝑓, Ω). In each iteration of the algorithm, the new

vector𝑥_𝑘+1is calculated according to the following iterative

scheme. Given the current vector𝑥_𝑘, compute𝑟(𝑥_𝑘) = 𝑥_𝑘−

𝑃Ω(𝑥𝑘− 𝑓(𝑥𝑘)), if𝑟(𝑥𝑘) = 0, stop; otherwise, compute

𝑧_𝑘 = 𝑥_𝑘− 𝜂_𝑘𝑟 (𝑥_𝑘) , (3)

where𝜂_𝑘= 𝑟𝑚𝑘_and𝑚

𝑘being the smallest nonnegative integer

𝑚satisfying

⟨𝑓 (𝑥_𝑘− 𝑟𝑚𝑟 (𝑥_𝑘)) , 𝑟 (𝑥_𝑘_{)⟩ ≥ 𝛿󵄩󵄩󵄩󵄩𝑟 (𝑥}_𝑘_{)󵄩󵄩󵄩󵄩}2, (4) and then compute

𝑥_𝑘+1= 𝑃_Ω∩𝐻_𝑘(𝑥_𝑘) , (5)

where𝐻_𝑘= {𝑥 ∈ 𝑅𝑛 | ⟨𝑥 − 𝑧_𝑘, 𝑓(𝑧_𝑘)⟩ ≤ 0}.

On the other hand, Nadezhkina and Takahashi [11] got

𝑥_𝑘+1by the following iterative formula:

𝑥_𝑘+1= 𝛼_𝑘𝑥_𝑘+ (1 − 𝛼_𝑘) 𝑆𝑃_Ω(𝑥_𝑘− 𝜆_𝑘𝑓 (𝑥_𝑘)) , (6)

where{𝛼_𝑘}∞_𝑘=0is a sequence in(0, 1), {𝜆_𝑘}∞_𝑘=0 is a sequence,

and𝑆 : Ω → Ω is a nonexpansive mapping. Denoting the

fixed points set of𝑆by𝐹(𝑆)and assuming𝐹(𝑆)∩Ω∗ ̸= 0, they

proved that the sequence{𝑥_𝑘}∞_𝑘=0converges weakly to some

𝑥∗ ∈ 𝐹(𝑆) ∩ Ω∗.

Motivated and inspired by the extragradient methods in

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and analyze the weak converge property of three sequences generated by our method.

The rest of this paper is organized as follows. In Section2,

we give some preliminaries and basic results. In Section3, we

present an extragradient algorithm and then discuss the weak convergence of the sequences generated by the algorithm. In

Section4, we modify the extragradient algorithm and give its

convergence analysis.

2. Preliminary and Basic Results

Let 𝐻 be a real Hilbert space with ⟨𝑥, 𝑦⟩ denoting the

inner product of the vectors𝑥, 𝑦. Weak converge and strong

converge of the sequence{𝑥_𝑘}∞_𝑘=0to a point𝑥are denoted by

𝑥_𝑘 ⇀ 𝑥and𝑥_𝑘 → 𝑥, respectively. Identity mapping fromΩ

to itself is denoted by𝐼.

For some vector𝑥 ∈ 𝐻, the orthogonal projection of𝑥

ontoΩ, denoted by𝑃_Ω(𝑥), is defined as

𝑃_Ω(𝑥) :=arg min{󵄩󵄩󵄩󵄩𝑦 − 𝑥󵄩󵄩󵄩󵄩 | 𝑦 ∈ Ω}. (7) The following lemma states some well-known properties of the orthogonal projection operator.

Lemma 1. One has

(1) ⟨𝑥 − 𝑃Ω(𝑥) , 𝑃Ω(𝑥) − 𝑦 ⟩ ≥ 0, ∀ 𝑥 ∈ 𝑅𝑛, ∀ 𝑦 ∈ Ω. (8) (2) 󵄩󵄩󵄩󵄩𝑃Ω(𝑥) − 𝑃Ω(𝑦)󵄩󵄩󵄩󵄩 ≤󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩, ∀𝑥, 𝑦 ∈ 𝑅𝑛. (9) (3) 󵄩󵄩󵄩󵄩𝑃Ω(𝑥) − 𝑦󵄩󵄩󵄩󵄩2≤ 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩2− 󵄩󵄩󵄩󵄩𝑥 − 𝑃Ω(𝑥)󵄩󵄩󵄩󵄩2, ∀ 𝑥 ∈ 𝑅𝑛, ∀ 𝑦 ∈ Ω. (10) (4) 󵄩󵄩󵄩󵄩𝑃Ω(𝑥) − 𝑃Ω(𝑦)󵄩󵄩󵄩󵄩2 ≤ 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩2 − 󵄩󵄩󵄩󵄩𝑃Ω(𝑥) − 𝑥 + 𝑦 − 𝑃Ω(𝑦)󵄩󵄩󵄩󵄩2, ∀ 𝑥, 𝑦 ∈ 𝑅𝑛. (11)

A mapping𝑓is called monotone if

⟨𝑥 − 𝑦, 𝑓 (𝑥) − 𝑓 (𝑦)⟩ ≥ 0, ∀ 𝑥, 𝑦 ∈ Ω. (12)

A mapping𝑓is called Lipschitz continuous, if there exists an

𝐿 ≥ 0, such that

󵄩󵄩󵄩󵄩𝑓(𝑥) − 𝑓(𝑦)󵄩󵄩󵄩󵄩 ≤ 𝐿󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩, ∀𝑥,𝑦 ∈ Ω. (13)

The graph of𝑓, denoted by𝐺(𝑓), is defined by

𝐺 (𝑓) := {(𝑥, 𝑦 ) ∈ Ω × Ω | 𝑦 = 𝑓 (𝑥)} . (14)

A mapping𝑆 : Ω → Ω is called nonexpansive if

󵄩󵄩󵄩󵄩𝑆(𝑥) − 𝑆(𝑦 )󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩, ∀𝑥,𝑦 ∈ Ω, (15)

and the fixed point set of a mapping𝑆, denoted by𝐹(𝑆), is

defined by

𝐹 (𝑆) := {𝑥 ∈ Ω | 𝑆 (𝑥) = 𝑥} . (16)

We denote the normal cone ofΩatV∈ Ωby

𝑁_Ω(V) := {𝑤 ∈ 𝐻 | ⟨V− 𝑢, 𝑤⟩ ≥ 0, 𝑢 ∈ Ω} , (17)

and define the function𝑇(V)as

𝑇 (V) := {𝑓 (V) + 𝑁Ω(V) if V∈ Ω,

0 if V∉ Ω. (18)

Then𝑇is maximal monotone. It is well known that0 ∈ 𝑇(V),

if and only ifV ∈ Ω∗. For more details, see, for example, [9]

and references therein. The following lemma is established in Hilbert space and is well known as Opial condition.

Lemma 2. For any sequence {𝑥_𝑘}∞_𝑘=0 ⊂ 𝐻 that converges

weakly to𝑥(𝑥_𝑘⇀ 𝑥), one has

lim inf

𝑘 → ∞ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥󵄩󵄩󵄩󵄩 <lim inf𝑘 → ∞ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑦󵄩󵄩󵄩󵄩 , ∀𝑦 ̸= 𝑥. (19)

The next lemma is proposed in [10].

Lemma 3 (Demiclosedness principle). Let Ω be a closed,

convex subset of a real Hilbert space𝐻, and let𝑆 : Ω → 𝐻

be a nonexpansive mapping. Then𝐼−𝑆is demiclosed at𝑦 ∈ 𝐻;

that is, for any sequence{𝑥_𝑘}∞_𝑘=0⊂ Ω, such that𝑥_𝑘⇀ ̃𝑥, ̃𝑥 ∈ Ω

and(𝐼 − 𝑆)𝑥_𝑘 → 𝑦, one has(𝐼 − 𝑆)̃𝑥 = 𝑦.

3. An Algorithm and Its Convergence Analysis

In this section, we give our algorithm, and then discuss its convergence. First, we need the following definition.

Definition 4. For some vector𝑥 ∈ Ω, the projected residual

function is defined as

𝑟 (𝑥) := 𝑥 − 𝑃Ω(𝑥 − 𝑓 (𝑥)) . (20)

Obviously, we have that𝑥 ∈ Ω∗if and only if𝑟(𝑥) = 0. Now

we describe our algorithm.

Algorithm A. Step 0. Take𝛿 ∈ (0, 1), 𝛾 ∈ (0, 1), 𝑥₀∈ Ω, and

𝑘 = 0.

Step 1. For the current iterative point𝑥_𝑘∈ Ω, compute

𝑦_𝑘= 𝑃_Ω(𝑥_𝑘− 𝑓 (𝑥_𝑘)) , (21)

𝑧_𝑘= (1 − 𝜂_𝑘) 𝑥_𝑘+ 𝜂_𝑘𝑦_𝑘, (22)

where𝜂_𝑘 = 𝛾𝑛𝑘_and𝑛

𝑘being the smallest nonnegative integer

𝑛satisfying ⟨𝑓 (𝑥_𝑘− 𝛾𝑛_{𝑟 (𝑥} 𝑘)) , 𝑟 (𝑥𝑘)⟩ ≥ 𝛿󵄩󵄩󵄩󵄩𝑟 (𝑥𝑘)󵄩󵄩󵄩󵄩2. (23) Compute 𝑡𝑘= 𝑃𝐻𝑘∩Ω(𝑥𝑘) , (24) 𝑥_𝑘+1= 𝛼_𝑘𝑥_𝑘+ (1 − 𝛼_𝑘) 𝑆𝑡_𝑘, (25) where{𝛼_𝑘} ⊂ (𝑎, 𝑏) (𝑎, 𝑏 ∈ (0, 1)), 𝐻_𝑘 = {𝑥 ∈ Ω | ⟨𝑥 −

𝑧𝑘, 𝑓(𝑧𝑘)⟩ ≤ 0}, and𝑆 : Ω → Ω is a nonexpansive mapping.

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Remark 5. The iterative point𝑡_𝑘 is well computed in

Algo-rithm A according to [3] and can be interpreted as follows: if

(23) is well defined, then𝑡_𝑘can be derived by the following

iterative scheme: compute

𝑥_𝑘= 𝑃_𝐻_𝑘(𝑥_𝑘) , 𝑡_𝑘= 𝑃_𝐻_𝑘_∩Ω(𝑥_𝑘) . (26)

For more details, see [3,4].

Now we investigate the weak convergence property of our algorithm. First we recall the following result, which was

proposed by Schu [17].

Lemma 6. Let H be a real Hilbert space, let {𝛼_𝑘}∞_𝑘=0 ⊂

(𝑎, 𝑏) (𝑎, 𝑏 ∈ (0, 1))be a sequence of real number, and let

{V_𝑘}∞ 𝑘=0⊂ 𝐻, {𝑤𝑘}∞𝑘=0⊂ 𝐻, such that lim sup 𝑘 → ∞ 󵄩󵄩󵄩󵄩V𝑘󵄩󵄩󵄩󵄩 ≤ 𝑐, lim sup 𝑘 → ∞ 󵄩󵄩󵄩󵄩𝑤𝑘󵄩󵄩󵄩󵄩 ≤ 𝑐, lim sup 𝑘 → ∞ 󵄩󵄩󵄩󵄩𝛼𝑘V𝑘+ (1 − 𝛼𝑘) 𝑤𝑘󵄩󵄩󵄩󵄩 = 𝑐, (27)

for some𝑐 ≥ 0. Then one has

lim

𝑘 → ∞󵄩󵄩󵄩󵄩V𝑘− 𝑤𝑘󵄩󵄩󵄩󵄩 = 0. (28)

The following theorem is crucial in proving the

bound-ness of the sequence{𝑥_𝑘}∞_𝑘=0.

Theorem 7. LetΩbe a nonempty, closed, and convex subset of

𝐻, let𝑓be a monotone and𝐿-Lipschitz continuous mapping,

𝐹(𝑆) ∩ Ω∗ _{̸= 0}_{, and} _𝑥∗ _{∈ Ω}∗_{. Then for any sequence}

{𝑥_𝑘}∞

𝑘=0, {𝑡𝑘}∞𝑘=0generated by Algorithm A, one has

󵄩󵄩󵄩󵄩𝑡𝑘− 𝑥∗󵄩󵄩󵄩󵄩2≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2− 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2

+ 2𝜂𝑘⟨𝑟 (𝑥𝑘) , 𝑓 (𝑧𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2

⟨𝑧_𝑘− 𝑡_𝑘, 𝑓 (𝑧_𝑘)⟩ . (29)

Proof. Letting𝑥 = 𝑥_𝑘and𝑦 = 𝑥∗. It follows from Lemma1

(10) that 󵄩󵄩󵄩󵄩 󵄩𝑃𝐻𝑘∩Ω(𝑥𝑘) − 𝑥 ∗󵄩󵄩󵄩󵄩 󵄩2≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2− 󵄩󵄩󵄩󵄩󵄩𝑥𝑘− 𝑃𝐻𝑘∩Ω(𝑥𝑘)󵄩󵄩󵄩󵄩󵄩 2 , (30) that is, 󵄩󵄩󵄩󵄩𝑡𝑘− 𝑥∗󵄩󵄩󵄩󵄩2≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2− 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2. (31)

From (20)–(23) in Algorithm A, we get⟨𝑥_𝑘− 𝑧_𝑘, 𝑓(𝑧_𝑘)⟩ > 0,

which means𝑥_𝑘 ∉ 𝐻_𝑘. So, by the definition of the projection

operator and [3], we obtain

𝑥_𝑘= 𝑃_𝐻_𝑘(𝑥_𝑘) = 𝑥_𝑘−⟨𝑥𝑘− 𝑧𝑘, 𝑓 (𝑧𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2 𝑓 (𝑧_𝑘) = 𝑥_𝑘−𝜂𝑘⟨𝑟 (𝑥𝑘) , 𝑓 (𝑧𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2 𝑓 (𝑧_𝑘) . (32)

Substituting (32) into (31), we have

󵄩󵄩󵄩󵄩𝑡𝑘− 𝑥∗󵄩󵄩󵄩󵄩2≤󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩󵄩𝑥𝑘− 𝜂_𝑘⟨𝑟(𝑥_𝑘), 𝑓(𝑧_𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2 𝑓(𝑧_𝑘) − 𝑥∗󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩󵄩 2 −󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩󵄩𝑥𝑘− 𝜂_𝑘⟨𝑟(𝑥_𝑘), 𝑓(𝑧_𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2 𝑓(𝑧_𝑘) − 𝑡_𝑘󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩󵄩 2 = 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2− 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2 − 2𝜂𝑘⟨𝑟 (𝑥𝑘) , 𝑓 (𝑧𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2 ⟨𝑡_𝑘− 𝑥∗, 𝑓 (𝑧_𝑘)⟩ = 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2− 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2 + 2𝜂𝑘⟨𝑟 (𝑥𝑘) , 𝑓 (𝑧𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2 ⟨𝑧𝑘− 𝑡𝑘, 𝑓 (𝑧𝑘)⟩ + 2𝜂𝑘⟨𝑟 (𝑥𝑘) , 𝑓 (𝑧𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2 ⟨𝑥∗− 𝑧_𝑘, 𝑓 (𝑧_𝑘)⟩ . (33)

Since𝑓is monotone, connecting with (1), we obtain

⟨𝑧𝑘− 𝑥∗, 𝑓 (𝑧𝑘)⟩ ≥ 0. (34) Thus 󵄩󵄩󵄩󵄩𝑡𝑘− 𝑥∗󵄩󵄩󵄩󵄩2≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2− 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2 + 2𝜂𝑘⟨𝑟 (𝑥𝑘) , 𝑓 (𝑧𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2 ⟨𝑧_𝑘− 𝑡_𝑘, 𝑓 (𝑧_𝑘)⟩ , (35)

which completes the proof.

Theorem 8. LetΩbe a nonempty, closed, and convex subset of

𝐻,𝑓be a monotone and𝐿-Lipschitz continuous mapping, and

𝐹(𝑆) ∩ Ω∗ _{̸= 0}_{. Then for any sequence}_{𝑥

𝑘}∞𝑘=0, {𝑦𝑘}∞𝑘=0, {𝑡𝑘}∞𝑘=0

generated by Algorithm A, one has

󵄩󵄩󵄩󵄩𝑥𝑘+1− 𝑥∗󵄩󵄩󵄩󵄩2≤󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2− (1 − 𝛼𝑘)(󵄩󵄩󵄩󵄩𝑓(𝑧𝜂𝑘𝛿 𝑘)󵄩󵄩󵄩󵄩) 2 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑦𝑘󵄩󵄩󵄩󵄩4, 󵄩󵄩󵄩󵄩𝑥𝑘+1− 𝑥∗󵄩󵄩󵄩󵄩2≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2−1 − 𝛼₂ 𝑘󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2. (36) Furthermore, lim 𝑘 → ∞󵄩󵄩󵄩󵄩𝑥𝑘− 𝑦𝑘󵄩󵄩󵄩󵄩 = 0, lim 𝑘 → ∞󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩 = 0, lim 𝑘 → ∞󵄩󵄩󵄩󵄩𝑦𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩 = 0. (37)

(4)

Proof. Using (22), we have 𝜂_𝑘⟨𝑟 (𝑥_𝑘) , 𝑓 (𝑧_𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2 ⟨𝑧𝑘− 𝑥𝑘, 𝑓 (𝑧𝑘)⟩ = 𝜂𝑘⟨𝑟 (𝑥𝑘) , 𝑓 (𝑧𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2 ⟨−𝜂_𝑘𝑟 (𝑥_𝑘) , 𝑓 (𝑧_𝑘)⟩ = −(𝜂𝑘⟨𝑟 (𝑥_{󵄩󵄩󵄩󵄩𝑓(𝑧}𝑘) , 𝑓 (𝑧𝑘)⟩ 𝑘)󵄩󵄩󵄩󵄩 ) 2 . (38)

By the Cauchy-Schwarz inequality,

2𝜂𝑘⟨𝑟 (𝑥𝑘) , 𝑓 (𝑧𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2 ⟨𝑥_𝑘− 𝑡_𝑘, 𝑓 (𝑧_𝑘)⟩ ≤ (𝜂𝑘⟨𝑟 (𝑥_{󵄩󵄩󵄩󵄩𝑓(𝑧}𝑘) , 𝑓 (𝑧𝑘)⟩ 𝑘)󵄩󵄩󵄩󵄩 ) 2 + 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2. (39) Hence, by (23) 󵄩󵄩󵄩󵄩𝑡𝑘− 𝑥∗󵄩󵄩󵄩󵄩2≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2− 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2 − (𝜂𝑘⟨𝑟 (𝑥_{󵄩󵄩󵄩󵄩𝑓(𝑧}𝑘) , 𝑓 (𝑧𝑘)⟩ 𝑘)󵄩󵄩󵄩󵄩 ) 2 + 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2 = 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2− (𝜂𝑘⟨𝑟 (𝑥󵄩󵄩󵄩󵄩𝑓(𝑧𝑘) , 𝑓 (𝑧𝑘)⟩ 𝑘)󵄩󵄩󵄩󵄩 ) 2 ≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2− (󵄩󵄩󵄩󵄩𝑓(𝑧𝜂𝑘𝛿 𝑘)󵄩󵄩󵄩󵄩) 2 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑦𝑘󵄩󵄩󵄩󵄩4. (40) Then we have 󵄩󵄩󵄩󵄩𝑥𝑘+1− 𝑥∗󵄩󵄩󵄩󵄩2= 󵄩󵄩󵄩󵄩𝛼𝑘𝑥𝑘+ (1 − 𝛼𝑘)𝑆𝑡𝑘− 𝑥∗󵄩󵄩󵄩󵄩2 = 󵄩󵄩󵄩󵄩𝛼𝑘(𝑥𝑘− 𝑥∗) + (1 − 𝛼𝑘)(𝑆𝑡𝑘− 𝑥∗)󵄩󵄩󵄩󵄩2 ≤ 𝛼_𝑘󵄩󵄩󵄩󵄩𝑥_𝑘− 𝑥∗󵄩󵄩󵄩󵄩2+ (1 − 𝛼_𝑘_{) 󵄩󵄩󵄩󵄩𝑡}_𝑘− 𝑥∗󵄩󵄩󵄩󵄩2 ≤ 𝛼_𝑘󵄩󵄩󵄩󵄩𝑥_𝑘− 𝑥∗󵄩󵄩󵄩󵄩2+ (1 − 𝛼_𝑘_{) 󵄩󵄩󵄩󵄩𝑥}_𝑘− 𝑥∗󵄩󵄩󵄩󵄩2 − (1 − 𝛼_𝑘) ( 𝜂𝑘𝛿 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩) 2 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑦𝑘󵄩󵄩󵄩󵄩4 = 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2−(1 − 𝛼𝑘)(󵄩󵄩󵄩󵄩𝑓(𝑧𝜂𝑘𝛿 𝑘)󵄩󵄩󵄩󵄩) 2 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑦𝑘󵄩󵄩󵄩󵄩4 ≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2, (41)

where the first inequation follows from that𝑆is a

nonexpan-sive mapping.

That means{𝑥_𝑘}∞_𝑘=0 is bounded, and so as{𝑧_𝑘}∞_𝑘=0. Since

𝑓is continuous; namely, there exists a constant𝑀 > 0, s.t.

‖𝑓(𝑧_𝑘)‖ ≤ 𝑀,for all𝑘, we yet have

󵄩󵄩󵄩󵄩𝑥𝑘+1− 𝑥∗󵄩󵄩󵄩󵄩2≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2− (1 − 𝛼𝑘) (_𝑀𝛿) 2

𝜂2_𝑘󵄩󵄩󵄩󵄩𝑥_𝑘− 𝑦_𝑘󵄩󵄩󵄩󵄩4.

(42)

So we know that there exists𝜉 ≥ 0,lim_{𝑘 → ∞}‖𝑥_𝑘− 𝑥∗‖ = 𝜉,

and hence

lim

𝑘 → ∞𝜂𝑘󵄩󵄩󵄩󵄩𝑥𝑘− 𝑦𝑘󵄩󵄩󵄩󵄩 = 0, (43)

which implies that lim_{𝑘 → ∞}‖𝑥_𝑘− 𝑦_𝑘‖ = 0 or lim_{𝑘 → ∞}𝜂_𝑘 = 0.

If lim_{𝑘 → ∞}‖𝑥_𝑘− 𝑦_𝑘‖ = 0, we get the conclusion.

If lim_{𝑘 → ∞}𝜂_𝑘 = 0, we can deduce that the inequality (23)

in Algorithm A is not satisfied for𝑛_𝑘− 1; that is, there exists

𝑘₀, for all𝑘 ≥ 𝑘₀,

⟨𝑓 (𝑥_𝑘− 𝛾−1𝜂_𝑘𝑟 (𝑥_𝑘)) , 𝑟 (𝑥_𝑘_{)⟩ < 𝛿󵄩󵄩󵄩󵄩𝑟 (𝑥}_𝑘_{)󵄩󵄩󵄩󵄩}2. (44)

Applying (8) by setting𝑥 = 𝑥_𝑘− 𝑓(𝑥_𝑘), 𝑦= 𝑥_𝑘leads to

⟨𝑥𝑘−𝑓 (𝑥𝑘)−𝑃Ω(𝑥𝑘− 𝑓 (𝑥𝑘)) , 𝑃Ω(𝑥𝑘− 𝑓 (𝑥𝑘)) − 𝑥𝑘⟩ ≥ 0.

(45) Therefore

⟨𝑓 (𝑥𝑘), 𝑟 (𝑥𝑘)⟩≥󵄩󵄩󵄩󵄩𝑟 (𝑥𝑘)󵄩󵄩󵄩󵄩2 as𝑟 (𝑥𝑘)=𝑥𝑘−𝑃Ω(𝑥𝑘−𝑓 (𝑥𝑘)) .

(46)

Passing onto the limit in (44), (46), we get𝛿lim_{𝑘 → ∞}‖𝑥_𝑘−

𝑦_𝑘‖ ≥ lim_{𝑘 → ∞}‖𝑥_𝑘 − 𝑦_𝑘‖, since 𝛿 ∈ (0, 1), we obtain lim_{𝑘 → ∞}‖𝑥_𝑘− 𝑦_𝑘‖ = 0.

On the other hand, using Cauchy-Schwarz inequality again, we have 2𝜂𝑘⟨𝑟 (𝑥𝑘) , 𝑓 (𝑧𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2 ⟨𝑥_𝑘− 𝑡_𝑘, 𝑓 (𝑧_𝑘)⟩ ≤ 2(𝜂𝑘⟨𝑟 (𝑥_{󵄩󵄩󵄩󵄩𝑓(𝑧}𝑘) , 𝑓 (𝑧𝑘)⟩ 𝑘)󵄩󵄩󵄩󵄩 ) 2 +1₂ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2. (47) Therefore, 󵄩󵄩󵄩󵄩𝑡𝑘− 𝑥∗󵄩󵄩󵄩󵄩2 ≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2− 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2− 2(𝜂𝑘⟨𝑟 (𝑥󵄩󵄩󵄩󵄩𝑓(𝑧𝑘) , 𝑓 (𝑧𝑘)⟩ 𝑘)󵄩󵄩󵄩󵄩 ) 2 + 2(𝜂𝑘⟨𝑟 (𝑥_{󵄩󵄩󵄩󵄩𝑓(𝑧}𝑘) , 𝑓 (𝑧𝑘)⟩ 𝑘)󵄩󵄩󵄩󵄩 ) 2 +1₂ 󵄩󵄩󵄩󵄩𝑥_𝑘− 𝑡_𝑘󵄩󵄩󵄩󵄩2 ≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2−1₂ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2. (48) Then we have 󵄩󵄩󵄩󵄩𝑥𝑘+1− 𝑥∗󵄩󵄩󵄩󵄩2= 󵄩󵄩󵄩󵄩𝛼𝑘𝑥𝑘+ (1 − 𝛼𝑘)𝑆𝑡𝑘− 𝑥∗󵄩󵄩󵄩󵄩2 ≤ 𝛼𝑘󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2+ (1 − 𝛼𝑘) 󵄩󵄩󵄩󵄩𝑡𝑘− 𝑥∗󵄩󵄩󵄩󵄩2 ≤ 𝛼_𝑘󵄩󵄩󵄩󵄩𝑥_𝑘− 𝑥∗󵄩󵄩󵄩󵄩2+ (1 − 𝛼_𝑘_{) 󵄩󵄩󵄩󵄩𝑥}_𝑘− 𝑥∗󵄩󵄩󵄩󵄩2 −1 − 𝛼𝑘 2 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩 2 = 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2−1 − 𝛼₂ 𝑘󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2. (49)

(5)

Noting that𝛼_𝑘 ∈ (𝑎, 𝑏) (𝑎, 𝑏 ∈ (0, 1)), it easily follows that

0 < (1 − 𝑏)/2 < (1 − 𝛼_𝑘)/2 < (1 − 𝑎)/2 < 1, which implies that lim

𝑘 → ∞󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩 = 0. (50)

By the triangle inequality, we have

󵄩󵄩󵄩󵄩𝑦𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩𝑦𝑘− 𝑥𝑘+ 𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩𝑦𝑘− 𝑥𝑘󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩.

(51)

Passing onto the limit in (51), we conclude

lim

𝑘 → ∞󵄩󵄩󵄩󵄩𝑦𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩 = 0. (52)

The proof is complete.

Theorem 9. Let Ω be a nonempty, closed, and convex

subset of 𝐻, let 𝑓 be a monotone and 𝐿-Lipschitz

con-tinuous mapping, and 𝐹(𝑆) ∩ Ω∗ ̸= 0. Then the sequences

{𝑥_𝑘}∞

𝑘=0, {𝑦𝑘}∞𝑘=0, {𝑡𝑘}∞𝑘=0 generated by Algorithm A converge

weakly to the same point 𝑥∗ ∈ 𝐹(𝑆) ∩ Ω∗, where 𝑥∗ =

lim_{𝑘 → ∞}𝑃_{𝐹(𝑆)∩Ω}∗(𝑥_𝑘).

Proof. By Theorem8, we know that{𝑥_𝑘}∞_𝑘=0 is bound, which

implies that there exists a subsequence{𝑥_𝑘_𝑖}∞_𝑖=0of{𝑥_𝑘}∞_𝑘=0that

converges weakly to some points𝑥∗ ∈ 𝐹(𝑆) ∩ Ω∗.

First, we investigate some details of𝑥∗ ∈ 𝐹(𝑆).

Letting𝑥󸀠∈ 𝐹(𝑆) ∩ Ω∗, since𝑆is nonexpansive mapping,

from (29) we have

󵄩󵄩󵄩󵄩

󵄩𝑆𝑡𝑘− 𝑥󸀠󵄩󵄩󵄩󵄩_{󵄩 ≤}󵄩󵄩󵄩󵄩_󵄩𝑡𝑘− 𝑥󸀠󵄩󵄩󵄩󵄩_{󵄩 ≤}󵄩󵄩󵄩󵄩_󵄩𝑥𝑘− 𝑥󸀠󵄩󵄩󵄩󵄩_{󵄩 .} (53)

Passing onto the limit in (53), we obtain

lim 𝑘 → ∞󵄩󵄩󵄩󵄩󵄩𝑆𝑡𝑘− 𝑥 󸀠󵄩󵄩󵄩󵄩 󵄩 ≤ 𝜉. (54) Then by (25) we have lim 𝑘 → ∞󵄩󵄩󵄩󵄩󵄩𝛼𝑘(𝑥𝑘− 𝑥 󸀠_{) + (1 − 𝛼} 𝑘) (𝑆𝑡𝑘− 𝑥󸀠)󵄩󵄩󵄩󵄩󵄩 = lim 𝑘 → ∞󵄩󵄩󵄩󵄩󵄩𝑥𝑘+1− 𝑥 󸀠󵄩󵄩󵄩󵄩 󵄩 = 𝜉. (55)

From Lemma6, it follows that

lim

𝑘 → ∞󵄩󵄩󵄩󵄩𝑆𝑡𝑘− 𝑥𝑘󵄩󵄩󵄩󵄩 = 0. (56)

By the triangle inequality, we have

󵄩󵄩󵄩󵄩𝑆𝑥𝑘− 𝑥𝑘󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩𝑆𝑥𝑘− 𝑆𝑡𝑘󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑆𝑡𝑘− 𝑥𝑘󵄩󵄩󵄩󵄩

≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑆𝑡𝑘− 𝑥𝑘󵄩󵄩󵄩󵄩,

(57)

and then passing onto the limit in (57), we deduce that

lim

𝑘 → ∞󵄩󵄩󵄩󵄩𝑆𝑥𝑘− 𝑥𝑘󵄩󵄩󵄩󵄩 = 0, (58)

which imply that𝑥∗∈ 𝐹(𝑆)by Lemma3.

Second, we describe the details of𝑥∗∈ Ω∗.

Since𝑥_𝑘_𝑖 ⇀ 𝑥∗, using Theorem8we claim that𝑡_𝑘_𝑖 ⇀ 𝑥∗

and𝑦_𝑘_𝑖 ⇀ 𝑥∗.

Letting(V, 𝑢) ∈ 𝐺(𝑇), we have

𝑢 ∈ 𝑇 (V) = 𝑓 (V) + 𝑁_Ω(V) , 𝑢 − 𝑓 (V) ∈ 𝑁_Ω(V) , (59) thus,

⟨V− 𝑥∗, 𝑢 − 𝑓 (V)⟩ ≥ 0, ∀ 𝑥∗∈ Ω. (60)

Applying (8) by letting𝑥 = 𝑥_𝑘− 𝑓(𝑥_𝑘), 𝑦=V, we have

⟨𝑥_𝑘− 𝑓 (𝑥_𝑘) − 𝑃_Ω(𝑥_𝑘− 𝑓 (𝑥_𝑘)) , 𝑃_Ω(𝑥_𝑘− 𝑓 (𝑥_𝑘)) −V⟩ ≥ 0,

(61) that is,

⟨𝑥_𝑘− 𝑓 (𝑥_𝑘) − 𝑦_𝑘, 𝑦_𝑘−V⟩ ≥ 0. (62)

Note that𝑢 − 𝑓(V) ∈ 𝑁_Ω(V)and𝑡_𝑘∈ Ω, then

⟨V− 𝑡_𝑘_𝑖, 𝑢⟩ ≥ ⟨V− 𝑡_𝑘_𝑖, 𝑓 (V)⟩ ≥ ⟨V− 𝑡_𝑘_𝑖, 𝑓 (V)⟩ − ⟨𝑥_𝑘_𝑖− 𝑓 (𝑥_𝑘_𝑖) − 𝑦_𝑘_𝑖, 𝑦_𝑘_𝑖−V⟩ = ⟨V− 𝑡_𝑘_𝑖, 𝑓 (V) − 𝑓 (𝑡_𝑘_𝑖)⟩ + ⟨V− 𝑡_𝑘_𝑖, 𝑓 (𝑡_𝑘_𝑖)⟩ − ⟨V− 𝑦_𝑘_𝑖, 𝑦_𝑘_𝑖− 𝑥_𝑘_𝑖⟩ − ⟨V− 𝑦_𝑘_𝑖, 𝑓 (𝑥_𝑘_𝑖)⟩ ≥ ⟨V− 𝑡𝑘𝑖, 𝑓 (𝑡𝑘𝑖)⟩ − ⟨V− 𝑦𝑘𝑖, 𝑦𝑘𝑖− 𝑥𝑘𝑖⟩ − ⟨V− 𝑦_𝑘_𝑖, 𝑓 (𝑥_𝑘_𝑖)⟩ , (63)

where the last inequation follows from the monotone of𝑓.

Since𝑓is continuous, by (37) we have

lim

𝑖 → ∞𝑓 (𝑥𝑘𝑖) =_{𝑖 → ∞}lim𝑓 (𝑦𝑘𝑖) , _{𝑖 → ∞}lim𝑥𝑘𝑖 =_{𝑖 → ∞}lim𝑡𝑘𝑖. (64)

Passing onto the limit in (63), we obtain

⟨V− 𝑥∗, 𝑢⟩ ≥ 0. (65)

As𝑓is maximal monotone, we have 𝑥∗ ∈ 𝑓−1(0), which

implies that𝑥∗∈ Ω∗.

At last we show that such𝑥∗is unique.

Let{𝑥_𝑘_𝑗}∞_𝑗=0be another subsequence of{𝑥_𝑘}∞_𝑘=0, such that

𝑥_𝑘_𝑗 ⇀ 𝑥∗

Δ. Then we conclude that𝑥∗Δ ∈ 𝐹(𝑆) ∩ Ω∗. Suppose

𝑥∗_Δ ̸= 𝑥∗; by Lemma2we have lim 𝑘 → ∞󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥 ∗󵄩󵄩󵄩󵄩 =lim inf 𝑖 → ∞ 󵄩󵄩󵄩󵄩󵄩𝑥𝑘𝑖−𝑥 ∗󵄩󵄩󵄩󵄩 󵄩<lim inf𝑖 → ∞ 󵄩󵄩󵄩󵄩󵄩𝑥𝑘𝑖−𝑥 ∗ Δ󵄩󵄩󵄩󵄩_󵄩=_{𝑘 → ∞}lim 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗Δ󵄩󵄩󵄩󵄩 =lim inf 𝑗 → ∞ 󵄩󵄩󵄩󵄩󵄩󵄩𝑥𝑘𝑗−𝑥 ∗ Δ󵄩󵄩󵄩󵄩󵄩󵄩<lim inf_{𝑗 → ∞} 󵄩󵄩󵄩󵄩󵄩󵄩𝑥𝑘𝑗−𝑥 ∗󵄩󵄩󵄩󵄩 󵄩󵄩= lim 𝑘 → ∞󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥 ∗󵄩󵄩󵄩󵄩, (66)

which implies that lim_{𝑘 → ∞}‖𝑥_𝑘− 𝑥∗‖ < lim_{𝑘 → ∞}‖𝑥_𝑘− 𝑥∗‖,

and this is a contradiction. Thus,𝑥∗ = 𝑥∗_Δ, and the proof is

(6)

4. Further Study

In this section we propose an extension of Algorithm A, which is effective in practice. Similar to the investigation in

Section3, for the constant𝜏 > 0, we define a new projected

residual function as follows:

𝑟 (𝑥, 𝜏) := 𝑥 − 𝑃Ω(𝑥 − 𝜏𝑓 (𝑥)) . (67)

It is clear that the new projected residual function (67)

degenerates into (20) by setting𝜏 = 1.

Algorithm B. Step 0. Take𝛿 ∈ (0, 1), 𝛾 ∈ (0, 1), 𝜂₋₁ > 0, 𝜃 >

1, 𝑥₀∈ Ω, and𝑘 = 0.

Step 1. For the current iterative point𝑥_𝑘∈ Ω, compute

𝑦_𝑘 = 𝑃_Ω(𝑥_𝑘− 𝜏_𝑘𝑓 (𝑥_𝑘)) ,

𝑧_𝑘= (1 − 𝜂_𝑘) 𝑥_𝑘+ 𝜂_𝑘𝑦_𝑘, (68)

where 𝜏_𝑘 = min{𝜃𝜂_𝑘−1, 1}, 𝜂_𝑘 = 𝛾𝑛𝑘𝜏

𝑘 and 𝑛𝑘 being the

smallest nonnegative integer𝑛satisfying

⟨𝑓 (𝑥_𝑘− 𝛾𝑛𝜏_𝑘𝑟 (𝑥_𝑘, 𝜏_𝑘)) , 𝑟 (𝑥_𝑘, 𝜏_𝑘)⟩ ≥ 𝛿 𝜏_𝑘 󵄩󵄩󵄩󵄩𝑟(𝑥𝑘, 𝜏𝑘)󵄩󵄩󵄩󵄩 2_. (69) Compute 𝑡_𝑘= 𝑃_𝐻_𝑘_∩Ω(𝑥_𝑘) , 𝑥_𝑘+1= 𝛼_𝑘𝑥_𝑘+ (1 − 𝛼_𝑘) 𝑆𝑡_𝑘, (70) where{𝛼_𝑘} ⊂ (𝑎, 𝑏) (𝑎, 𝑏 ∈ (0, 1))and𝐻_𝑘 = {𝑥 ∈ Ω | ⟨𝑥 − 𝑧_𝑘, 𝑓(𝑧_𝑘)⟩ ≤ 0}.

Step 2. If‖𝑟(𝑥_𝑘, 𝜏_𝑘)‖ = 0, stop; otherwise go to Step 1.

At the rest of this section, we discuss the weak conver-gence property of Algorithm B.

Lemma 10. For any𝜏 > 0, one has

𝑥∗is the solution of VI(𝑓, Ω) ⇐⇒ 𝑥∗= 𝑃_Ω(𝑥∗− 𝜏𝑓 (𝑥∗)) .

(71) Therefore, solving variational inequality is equivalent to

finding a zero point of the projected residual function𝑟(∙, 𝜏).

Meanwhile we know that𝑟(𝑥, 𝜏)is a continuous function of

𝑥, as the projection mapping is nonexpansive.

Lemma 11. For any𝑥 ∈ 𝑅𝑛 and𝜏₁≥ 𝜏₂> 0, it holds that

󵄩󵄩󵄩󵄩𝑟(𝑥,𝜏1)󵄩󵄩󵄩󵄩 ≥󵄩󵄩󵄩󵄩𝑟(𝑥,𝜏2)󵄩󵄩󵄩󵄩 . (72) Theorem 12. LetΩbe a nonempty, closed, and convex subset

of𝐻, let𝑓be a monotone and𝐿-Lipschitz continuous mapping,

and𝐹(𝑆) ∩ Ω∗ ̸= 0. Then for any sequence {𝑥_𝑘}∞_𝑘=0, {𝑡_𝑘}∞_𝑘=0 generated by Algorithm B, one has

󵄩󵄩󵄩󵄩𝑡𝑘− 𝑥∗󵄩󵄩󵄩󵄩2≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2− 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2

+ 2𝜂𝑘⟨𝑟 (𝑥𝑘, 𝜏𝑘) , 𝑓 (𝑧𝑘)⟩ 󵄩󵄩󵄩󵄩𝑓(𝑧𝑘)󵄩󵄩󵄩󵄩2

⟨𝑧_𝑘− 𝑡_𝑘, 𝑓 (𝑧_𝑘)⟩ .

(73)

Proof. The proof of this theorem is similar to Theorem7, so

we omit it.

contin-uous mapping, and 𝐹(𝑆) ∩ Ω∗ ̸= 0. Then for any sequences

{𝑥_𝑘}∞

𝑘=0, {𝑦𝑘}∞𝑘=0, {𝑡𝑘}∞𝑘=0 generated by Algorithm B, one has

󵄩󵄩󵄩󵄩𝑥𝑘+1− 𝑥∗󵄩󵄩󵄩󵄩2≤ 󵄩󵄩󵄩󵄩𝑥𝑘− 𝑥∗󵄩󵄩󵄩󵄩2−1 − 𝛼₂ 𝑘󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩2. (74) Furthermore, lim 𝑘 → ∞󵄩󵄩󵄩󵄩𝑥𝑘− 𝑦𝑘󵄩󵄩󵄩󵄩 = 0, lim 𝑘 → ∞󵄩󵄩󵄩󵄩𝑥𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩 = 0, lim 𝑘 → ∞󵄩󵄩󵄩󵄩𝑦𝑘− 𝑡𝑘󵄩󵄩󵄩󵄩 = 0. (75)

Proof. The proof of this theorem is similar to the Theorem8.

The only difference is that (44) is substituted by

⟨𝑓 (𝑥_𝑘− 𝛾−1𝜂_𝑘𝑟 (𝑥_𝑘, 𝜏_𝑘)) , 𝑟 (𝑥_𝑘, 𝜏_𝑘)⟩ < 𝛿

𝜏_𝑘 󵄩󵄩󵄩󵄩𝑟(𝑥𝑘, 𝜏𝑘)󵄩󵄩󵄩󵄩

2_,

(76)

where (76) follows from Lemma11with𝜏₁ = 1and𝜏₂ = 𝜏_𝑘.

con-tinuous mapping, and 𝐹(𝑆) ∩ Ω∗ ̸= 0. Then the sequences

{𝑥_𝑘}∞_𝑘=0, {𝑦_𝑘}∞_𝑘=0, {𝑡_𝑘}∞_𝑘=0 generated by Algorithm B converge

weakly to the same point 𝑥∗ ∈ 𝐹(𝑆) ∩ Ω∗, where 𝑥∗ =

lim

𝑘 → ∞𝑃𝐹(𝑆)∩Ω∗(𝑥𝑘).

5. Conclusions

In this paper, we proposed an extension of the extragradient algorithm for solving monotone variational inequalities and established its weak convergence theorem. The Algorithm B is effective in practice. Meanwhile, we pointed out that the solution of our algorithm is also a fixed point of a given nonexpansive mapping.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant: 11171362) and the Funda-mental Research Funds for the central universities (Grant: CDJXS12101103). The authors thank the anonymous review-ers for their valuable comments and suggestions, which helped to improve the paper.

References

[1] R. W. Cottle, J.-S. Pang, and R. E. Stone,The Linear

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[2] G. M. Korpelevich, “The extragradient method for finding

saddle points and other problems,”Matecon, vol. 12, pp. 747–

756, 1976.

[3] M. V. Solodov and B. F. Svaiter, “A new projection method for

variational inequality problems,”SIAM Journal on Control and

Optimization, vol. 37, no. 3, pp. 765–776, 1999.

[4] M. V. Solodov, “Stationary points of bound constrained

mini-mization reformulations of complementarity problems,”Journal

of Optimization Theory and Applications, vol. 94, no. 2, pp. 449– 467, 1997.

[5] Y. J. Wang, N. H. Xiu, and J. Z. Zhang, “Modified extragradient method for variational inequalities and verification of solution

existence,”Journal of Optimization Theory and Applications, vol.

119, no. 1, pp. 167–183, 2003.

[6] W. Takahashi and M. Toyoda, “Weak convergence theorems for

nonexpansive mappings and monotone mappings,”Journal of

Optimization Theory and Applications, vol. 118, no. 2, pp. 417– 428, 2003.

[7] E. H. Zarantonello,Projections on Convex Sets in Hilbert Space

and Spectral Theory, Contributions to Nonlinear Functional Analysis, Academic press, New York, NY, USA, 1971.

[8] Z. Opial, “Weak convergence of the sequence of successive

approximations for nonexpansive mappings,” Bulletin of the

American Mathematical Society, vol. 73, pp. 591–597, 1967. [9] R. T. Rockafellar, “On the maximality of sums of nonlinear

monotone operators,”Transactions of the American

Mathemat-ical Society, vol. 149, pp. 75–88, 1970.

[10] F. E. Browder, “Fixed-point theorems for noncompact mappings

in Hilbert space,” Proceedings of the National Academy of

Sciences of the United States of America, vol. 53, pp. 1272–1276, 1965.

[11] N. Nadezhkina and W. Takahashi, “Weak convergence theorem by an extragradient method for nonexpansive mappings and

monotone mappings,” Journal of Optimization Theory and

Applications, vol. 128, no. 1, pp. 191–201, 2006.

[12] B. C. Eaves, “On the basic theorem of complementarity,”

Mathematical Programming, vol. 1, no. 1, pp. 68–75, 1971. [13] B. S. He and L. Z. Liao, “Improvements of some projection

methods for monotone nonlinear variational inequalities,”

Jour-nal of Optimization Theory and Applications, vol. 112, no. 1, pp. 111–128, 2002.

[14] Y. Censor, A. Gibali, and S. Reich, “The subgradient extragradi-ent method for solving variational inequalities in Hilbert space,”

Journal of Optimization Theory and Applications, vol. 148, no. 2, pp. 318–335, 2011.

[15] Y. Censor, A. Gibali, and S. Reich, “Two extensions of Kor-pelevich’s extragradient method for solving the variational inequality problem in Euclidean space,” Tech. Rep., 2010. [16] Y. J. Wang, N. H. Xiu, and C. Y. Wang, “Unified framework

of extragradient-type methods for pseudomonotone variational

inequalities,”Journal of Optimization Theory and Applications,

vol. 111, no. 3, pp. 641–656, 2001.

[17] J. Schu, “Weak and strong convergence to fixed points of

asymp-totically nonexpansive mappings,” Bulletin of the Australian

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