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

Abstract

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

Let Ω be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝑓∶Ω→Ω be a continuous mapping. The variational inequality problem, denoted by VI(𝑓,Ω), is to find a vector 𝑥^∗∈Ω, such that 𝑥−𝑥^∗,𝑓𝑥^∗≥0,∀𝑥∈Ω. (1) Throughout the paper, let Ω^∗ 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 vector 𝑥^∗∈Ω, such that 𝑥^∗≥0,𝑓𝑥^∗≥0,𝑓𝑥^∗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 mathematics 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 set Ω by the intersection of two sets related to VI(𝑓,Ω). In each iteration of the algorithm, the new vector 𝑥_𝑘+1 is calculated according to the following iterative scheme. Given the current vector 𝑥_𝑘, compute 𝑟(𝑥_𝑘)=𝑥_𝑘−𝑃_Ω(𝑥_𝑘−𝑓(𝑥_𝑘)), if 𝑟(𝑥_𝑘)=0, stop; otherwise, compute 𝑧_𝑘=𝑥_𝑘−𝜂_𝑘𝑟𝑥_𝑘, (3) where 𝜂_𝑘=𝑟^𝑚𝑘 and 𝑚_𝑘 being the smallest nonnegative integer 𝑚 satisfying 𝑓𝑥_𝑘−𝑟^𝑚𝑟𝑥_𝑘,𝑟𝑥_𝑘≥𝛿𝑟𝑥_𝑘², (4) and then compute 𝑥_𝑘+1=𝑃_{Ω∩𝐻𝑘}𝑥_𝑘, (5) where 𝐻_𝑘={𝑥∈𝑅^𝑛∣⟨𝑥−𝑧_𝑘,𝑓(𝑧_𝑘)⟩≤0}.

On the other hand, Nadezhkina and Takahashi [11] got 𝑥_𝑘+1 by the following iterative formula: 𝑥_𝑘+1=𝛼_𝑘𝑥_𝑘+1−𝛼_𝑘𝑆𝑃_Ω𝑥_𝑘−𝜆_𝑘𝑓𝑥_𝑘, (6) where {𝛼_𝑘}^∞_𝑘=0 is a sequence in (0,1),{𝜆_𝑘}^∞_𝑘=0 is a sequence, and 𝑆∶Ω→Ω is a nonexpansive mapping. Denoting the fixed points set of 𝑆 by 𝐹(𝑆) and assuming 𝐹(𝑆)∩Ω^∗≠∅, they proved that the sequence {𝑥_𝑘}^∞_𝑘=0 converges weakly to some 𝑥^∗∈𝐹(𝑆)∩Ω^∗.

Motivated and inspired by the extragradient methods in [2, 3], in this paper, we study further extragradient methods and analyze the weak converge property of three sequences generated by our method.

The rest of this paper is organized as follows. In Section 2, we give some preliminaries and basic results. In Section 3, we present an extragradient algorithm and then discuss the weak convergence of the sequences generated by the algorithm. In Section 4, 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 vectors 𝑥,𝑦. Weak converge and strong converge of the sequence {𝑥_𝑘}^∞_𝑘=0 to a point 𝑥 are denoted by 𝑥_𝑘⇀𝑥 and 𝑥_𝑘→𝑥, respectively. Identity mapping from Ω to itself is denoted by 𝐼.

For some vector 𝑥∈𝐻, the orthogonal projection of 𝑥 onto Ω, denoted by 𝑃_Ω(𝑥), is defined as 𝑃_Ω(𝑥)∶=argmin𝑦−𝑥∣𝑦∈Ω. (7) The following lemma states some well-known properties of the orthogonal projection operator.

Lemma 1. One has(1)𝑥−𝑃_Ω(𝑥),𝑃_Ω(𝑥)−𝑦≥0,∀𝑥∈𝑅^𝑛(2𝑃,∀𝑦∈Ω.)_Ω(𝑥)−𝑃_Ω𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝑅^𝑛.(3)𝑃_Ω(𝑥)−𝑦²≤𝑥−𝑦²−𝑥−𝑃_Ω(𝑥)²,∀𝑥∈𝑅^𝑛(4𝑃,∀𝑦∈Ω.)_Ω(𝑥)−𝑃_Ω𝑦²≤𝑥−𝑦²−𝑃_Ω(𝑥)−𝑥+𝑦−𝑃_Ω𝑦²,∀𝑥,𝑦∈𝑅^𝑛. (8)(9)(10)(11)

A mapping 𝑓 is called monotone if 𝑥−𝑦,𝑓(𝑥)−𝑓𝑦≥0,∀𝑥,𝑦∈Ω. (12) A mapping 𝑓 is called Lipschitz continuous, if there exists an 𝐿≥0, such that 𝑓(𝑥)−𝑓𝑦≤𝐿𝑥−𝑦,∀𝑥,𝑦∈Ω. (13) The graph of 𝑓, denoted by 𝐺(𝑓), is defined by 𝐺𝑓∶=𝑥,𝑦∈Ω×Ω∣𝑦=𝑓(𝑥). (14) A mapping 𝑆∶Ω→Ω is called nonexpansive if 𝑆(𝑥)−𝑆𝑦≤𝑥−𝑦,∀𝑥,𝑦∈Ω, (15) and the fixed point set of a mapping 𝑆, denoted by 𝐹(𝑆), is defined by 𝐹(𝑆)∶={𝑥∈Ω∣𝑆(𝑥)=𝑥}. (16) We denote the normal cone of Ω at 𝑣∈Ω by 𝑁_Ω(𝑣)∶={𝑤∈𝐻∣⟨𝑣−𝑢,𝑤⟩≥0,𝑢∈Ω}, (17) and define the function 𝑇(𝑣) as 𝑇(𝑣)𝑓(𝑣)∶=+𝑁_Ω(𝑣)if𝑣∈Ω,∅if𝑣∉Ω. (18) Then 𝑇 is maximal monotone. It is well known that 0∈𝑇(𝑣), if and only if 𝑣∈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 to 𝑥(𝑥_𝑘⇀𝑥), one has liminf_𝑘→∞𝑥_𝑘−𝑥<liminf_𝑘→∞𝑥_𝑘−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 space 𝐻, and let 𝑆∶Ω→𝐻 be a nonexpansive mapping. Then 𝐼−𝑆 is demiclosed at 𝑦∈𝐻; that is, for any sequence {𝑥_𝑘}^∞_𝑘=0⊂Ω, such that 𝑥_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 vector 𝑥∈Ω, the projected residual function is defined as 𝑟(𝑥)∶=𝑥−𝑃_Ω𝑥−𝑓(𝑥). (20) Obviously, we have that 𝑥∈Ω^∗ if and only if 𝑟(𝑥)=0. Now we describe our algorithm.

Algorithm A. Step  0. Take 𝛿∈(0,1),𝛾∈(0,1),𝑥₀∈Ω, and 𝑘=0.
Step 1. For the current iterative point 𝑥_𝑘∈Ω, compute 𝑦_𝑘=𝑃_Ω𝑥_𝑘−𝑓𝑥_𝑘,𝑧_𝑘=1−𝜂_𝑘𝑥_𝑘+𝜂_𝑘𝑦_𝑘, (21)(22) where 𝜂_𝑘=𝛾^𝑛𝑘 and 𝑛_𝑘 being the smallest nonnegative integer 𝑛 satisfying 𝑓𝑥_𝑘−𝛾^𝑛𝑟𝑥_𝑘,𝑟𝑥_𝑘≥𝛿𝑟𝑥_𝑘². (23) Compute 𝑡_𝑘=𝑃_{𝐻𝑘∩Ω}𝑥_𝑘,𝑥_𝑘+1=𝛼_𝑘𝑥_𝑘+1−𝛼_𝑘𝑆𝑡_𝑘, (24)(25) where {𝛼_𝑘}⊂(𝑎,𝑏)(𝑎,𝑏∈(0,1)),𝐻_𝑘={𝑥∈Ω∣⟨𝑥−𝑧_𝑘,𝑓(𝑧_𝑘)⟩≤0}, and 𝑆∶Ω→Ω is a nonexpansive mapping.
Step 2. If ‖𝑟(𝑥_𝑘+1)‖=0, stop; otherwise go to Step 1.

Remark 5. The iterative point 𝑡_𝑘 is well computed in Algorithm A according to [3] and can be interpreted as follows: if (23) is well defined, then 𝑡_𝑘 can be derived by the following iterative scheme: compute 𝑥_𝑘=𝑃_𝐻𝑘𝑥_𝑘,𝑡_𝑘=𝑃_{𝐻𝑘∩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,{𝑤_𝑘}^∞_𝑘=0⊂𝐻, such that limsup_𝑘→∞𝑣_𝑘≤𝑐,limsup_𝑘→∞𝑤_𝑘≤𝑐,limsup_𝑘→∞𝛼_𝑘𝑣_𝑘+1−𝛼_𝑘𝑤_𝑘=𝑐, (27) for some 𝑐≥0. Then one has lim_𝑘→∞𝑣_𝑘−𝑤_𝑘=0. (28)

The following theorem is crucial in proving the boundness of the sequence {𝑥_𝑘}^∞_𝑘=0.

Theorem 7. Let Ω be a nonempty, closed, and convex subset of 𝐻, let 𝑓 be a monotone and 𝐿-Lipschitz continuous mapping, 𝐹(𝑆)∩Ω^∗≠∅, and 𝑥^∗∈Ω^∗. Then for any sequence {𝑥_𝑘}^∞_𝑘=0,{𝑡_𝑘}^∞_𝑘=0 generated by Algorithm A, one has 𝑡_𝑘−𝑥^∗2≤𝑥_𝑘−𝑥^∗2−𝑥_𝑘−𝑡_𝑘²𝜂+2^𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓𝑧_𝑘²𝑧_𝑘−𝑡_𝑘,𝑓𝑧_𝑘. (29)

Proof. Letting 𝑥=𝑥_𝑘 and 𝑦=𝑥^∗. It follows from Lemma 1 (10) that 𝑃_{𝐻𝑘∩Ω}𝑥_𝑘−𝑥^∗2≤𝑥_𝑘−𝑥^∗2−𝑥_𝑘−𝑃_{𝐻𝑘∩Ω}𝑥_𝑘², (30) that is, 𝑡_𝑘−𝑥^∗2≤𝑥_𝑘−𝑥^∗2−𝑥_𝑘−𝑡_𝑘². (31) From (20)–(23) in Algorithm A, we get ⟨𝑥_𝑘−𝑧_𝑘,𝑓(𝑧_𝑘)⟩>0, which means 𝑥_𝑘∉𝐻_𝑘. So, by the definition of the projection operator and [3], we obtain 𝑥_𝑘=𝑃_𝐻𝑘𝑥_𝑘=𝑥_𝑘−𝑥_𝑘−𝑧_𝑘,𝑓𝑧_𝑘𝑓(𝑧_𝑘)²𝑓𝑧_𝑘=𝑥_𝑘−𝜂_𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓(𝑧_𝑘)²𝑓𝑧_𝑘. (32) Substituting (32) into (31), we have 𝑡_𝑘−𝑥^∗2≤𝑥_𝑘−𝜂_𝑘𝑟(𝑥_𝑘),𝑓(𝑧_𝑘)𝑓(𝑧_𝑘)²𝑓(𝑧_𝑘)−𝑥^∗2−𝑥_𝑘−𝜂_𝑘𝑟(𝑥_𝑘),𝑓(𝑧_𝑘)𝑓(𝑧_𝑘)²𝑓(𝑧_𝑘)−𝑡_𝑘²=𝑥_𝑘−𝑥^∗2−𝑥_𝑘−𝑡_𝑘²𝜂−2^𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓(𝑧_𝑘)²𝑡_𝑘−𝑥^∗,𝑓𝑧_𝑘=𝑥_𝑘−𝑥^∗2−𝑥_𝑘−𝑡_𝑘²𝜂+2^𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓(𝑧_𝑘)²𝑧_𝑘−𝑡_𝑘,𝑓𝑧_𝑘𝜂+2^𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓(𝑧_𝑘)²𝑥^∗−𝑧_𝑘,𝑓𝑧_𝑘. (33) Since 𝑓 is monotone, connecting with (1), we obtain 𝑧_𝑘−𝑥^∗,𝑓𝑧_𝑘≥0. (34) Thus 𝑡_𝑘−𝑥^∗2≤𝑥_𝑘−𝑥^∗2−𝑥_𝑘−𝑡_𝑘²𝜂+2^𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓𝑧_𝑘²𝑧_𝑘−𝑡_𝑘,𝑓𝑧_𝑘, (35) which completes the proof.

Theorem 8. Let Ω be a nonempty, closed, and convex subset of 𝐻, 𝑓 be a monotone and 𝐿-Lipschitz continuous mapping, and 𝐹(𝑆)∩Ω^∗≠∅. Then for any sequence {𝑥_𝑘}^∞_𝑘=0,{𝑦_𝑘}^∞_𝑘=0,{𝑡_𝑘}^∞_𝑘=0 generated by Algorithm A, one has 𝑥_𝑘+1−𝑥^∗2≤𝑥_𝑘−𝑥^∗2−1−𝛼_𝑘𝜂_𝑘𝛿𝑓𝑧_𝑘²𝑥_𝑘−𝑦_𝑘⁴,𝑥_𝑘+1−𝑥^∗2≤𝑥_𝑘−𝑥^∗2−1−𝛼_𝑘2𝑥_𝑘−𝑡_𝑘². (36) Furthermore, lim_𝑘→∞𝑥_𝑘−𝑦_𝑘=0,lim_𝑘→∞𝑥_𝑘−𝑡_𝑘=0,lim_𝑘→∞𝑦_𝑘−𝑡_𝑘=0. (37)

Proof. Using (22), we have 𝜂_𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓(𝑧_𝑘)²𝑧_𝑘−𝑥_𝑘,𝑓𝑧_𝑘=𝜂_𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓(𝑧_𝑘)²−𝜂_𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝜂=−^𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓𝑧_𝑘². (38) By the Cauchy-Schwarz inequality, 2𝜂_𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓(𝑧_𝑘)²𝑥_𝑘−𝑡_𝑘,𝑓𝑧_𝑘≤𝜂_𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓𝑧_𝑘²+𝑥_𝑘−𝑡_𝑘². (39) Hence, by (23) 𝑡_𝑘−𝑥^∗2≤𝑥_𝑘−𝑥^∗2−𝑥_𝑘−𝑡_𝑘²−𝜂_𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓𝑧_𝑘²+𝑥_𝑘−𝑡_𝑘²=𝑥_𝑘−𝑥^∗2−𝜂_𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓𝑧_𝑘²≤𝑥_𝑘−𝑥^∗2−𝜂_𝑘𝛿𝑓𝑧_𝑘²𝑥_𝑘−𝑦_𝑘⁴. (40) Then we have 𝑥_𝑘+1−𝑥^∗2=𝛼_𝑘𝑥_𝑘+(1−𝛼_𝑘)𝑆𝑡_𝑘−𝑥^∗2=𝛼_𝑘(𝑥_𝑘−𝑥^∗)+(1−𝛼_𝑘)(𝑆𝑡_𝑘−𝑥^∗)²≤𝛼_𝑘𝑥_𝑘−𝑥^∗2+1−𝛼_𝑘𝑡_𝑘−𝑥^∗2≤𝛼_𝑘𝑥_𝑘−𝑥^∗2+1−𝛼_𝑘𝑥_𝑘−𝑥^∗2−1−𝛼_𝑘𝜂_𝑘𝛿𝑓𝑧_𝑘²𝑥_𝑘−𝑦_𝑘⁴=𝑥_𝑘−𝑥^∗2−1−𝛼_𝑘𝜂_𝑘𝛿𝑓𝑧_𝑘²𝑥_𝑘−𝑦_𝑘⁴≤𝑥_𝑘−𝑥^∗2, (41) where the first inequation follows from that 𝑆 is a nonexpansive mapping.
That means {𝑥_𝑘}^∞_𝑘=0 is bounded, and so as {𝑧_𝑘}^∞_𝑘=0. Since 𝑓 is continuous; namely, there exists a constant 𝑀>0, s.t. ‖𝑓(𝑧_𝑘)‖≤𝑀,forall𝑘, we yet have 𝑥_𝑘+1−𝑥^∗2≤𝑥_𝑘−𝑥^∗2−1−𝛼_𝑘𝛿𝑀²𝜂²_𝑘𝑥_𝑘−𝑦_𝑘⁴. (42) So we know that there exists 𝜉≥0,lim_𝑘→∞‖𝑥_𝑘−𝑥^∗‖=𝜉, and hence lim_𝑘→∞𝜂_𝑘𝑥_𝑘−𝑦_𝑘=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 for 𝑛_𝑘−1; that is, there exists 𝑘₀, for all 𝑘≥𝑘₀, 𝑓𝑥_𝑘−𝛾⁻¹𝜂_𝑘𝑟𝑥_𝑘,𝑟𝑥_𝑘<𝛿𝑟𝑥_𝑘². (44) Applying (8) by setting 𝑥=𝑥_𝑘−𝑓(𝑥_𝑘),𝑦=𝑥_𝑘 leads to 𝑥_𝑘−𝑓𝑥_𝑘−𝑃_Ω𝑥_𝑘−𝑓𝑥_𝑘,𝑃_Ω𝑥_𝑘−𝑓𝑥_𝑘−𝑥_𝑘≥0. (45) Therefore 𝑓𝑥_𝑘,𝑟𝑥_𝑘≥𝑟𝑥_𝑘²as𝑟𝑥_𝑘=𝑥_𝑘−𝑃_Ω𝑥_𝑘−𝑓𝑥_𝑘. (46) Passing onto the limit in (44), (46), we get 𝛿lim_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 2𝜂_𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓𝑧_𝑘²𝑥_𝑘−𝑡_𝑘,𝑓𝑧_𝑘𝜂≤2^𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓𝑧_𝑘²+12𝑥_𝑘−𝑡_𝑘². (47) Therefore, 𝑡_𝑘−𝑥^∗2≤𝑥_𝑘−𝑥^∗2−𝑥_𝑘−𝑡_𝑘²𝜂−2^𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓𝑧_𝑘²𝜂+2^𝑘𝑟𝑥_𝑘,𝑓𝑧_𝑘𝑓𝑧_𝑘²+12𝑥_𝑘−𝑡_𝑘²≤𝑥_𝑘−𝑥^∗2−12𝑥_𝑘−𝑡_𝑘². (48) Then we have 𝑥_𝑘+1−𝑥^∗2=𝛼_𝑘𝑥_𝑘+(1−𝛼_𝑘)𝑆𝑡_𝑘−𝑥^∗2≤𝛼_𝑘𝑥_𝑘−𝑥^∗2+1−𝛼_𝑘𝑡_𝑘−𝑥^∗2≤𝛼_𝑘𝑥_𝑘−𝑥^∗2+1−𝛼_𝑘𝑥_𝑘−𝑥^∗2−1−𝛼_𝑘2𝑥_𝑘−𝑡_𝑘²=𝑥_𝑘−𝑥^∗2−1−𝛼_𝑘2𝑥_𝑘−𝑡_𝑘². (49) Noting that 𝛼_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→∞𝑥_𝑘−𝑡_𝑘=0. (50) By the triangle inequality, we have 𝑦_𝑘−𝑡_𝑘=𝑦_𝑘−𝑥_𝑘+𝑥_𝑘−𝑡_𝑘≤𝑦_𝑘−𝑥_𝑘+𝑥_𝑘−𝑡_𝑘. (51) Passing onto the limit in (51), we conclude lim_𝑘→∞𝑦_𝑘−𝑡_𝑘=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 continuous mapping, and 𝐹(𝑆)∩Ω^∗≠∅. Then the sequences {𝑥_𝑘}^∞_𝑘=0,{𝑦_𝑘}^∞_𝑘=0,{𝑡_𝑘}^∞_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 Theorem 8, we know that {𝑥_𝑘}^∞_𝑘=0 is bound, which implies that there exists a subsequence {𝑥_𝑘𝑖}^∞_𝑖=0 of {𝑥_𝑘}^∞_𝑘=0 that converges weakly to some points 𝑥^∗∈𝐹(𝑆)∩Ω^∗.
First, we investigate some details of 𝑥^∗∈𝐹(𝑆).
Letting 𝑥^′∈𝐹(𝑆)∩Ω^∗, since 𝑆 is nonexpansive mapping, from (29) we have 𝑆𝑡_𝑘−𝑥^′≤𝑡_𝑘−𝑥^′≤𝑥_𝑘−𝑥^′. (53) Passing onto the limit in (53), we obtain lim_𝑘→∞𝑆𝑡_𝑘−𝑥^′≤𝜉. (54) Then by (25) we have lim_𝑘→∞𝛼_𝑘𝑥_𝑘−𝑥^′+1−𝛼_𝑘𝑆𝑡_𝑘−𝑥^′=lim_𝑘→∞𝑥_𝑘+1−𝑥^′=𝜉. (55) From Lemma 6, it follows that lim_𝑘→∞𝑆𝑡_𝑘−𝑥_𝑘=0. (56) By the triangle inequality, we have 𝑆𝑥_𝑘−𝑥_𝑘≤𝑆𝑥_𝑘−𝑆𝑡_𝑘+𝑆𝑡_𝑘−𝑥_𝑘≤𝑥_𝑘−𝑡_𝑘+𝑆𝑡_𝑘−𝑥_𝑘, (57) and then passing onto the limit in (57), we deduce that lim_𝑘→∞𝑆𝑥_𝑘−𝑥_𝑘=0, (58) which imply that 𝑥^∗∈𝐹(𝑆)by Lemma 3.
Second, we describe the details of 𝑥^∗∈Ω^∗.
Since 𝑥_𝑘𝑖⇀𝑥^∗, using Theorem 8 we claim that 𝑡_𝑘𝑖⇀𝑥^∗ and 𝑦_𝑘𝑖⇀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 letting 𝑥=𝑥_𝑘−𝑓(𝑥_𝑘),𝑦=𝑣, we have 𝑥_𝑘−𝑓𝑥_𝑘−𝑃_Ω𝑥_𝑘−𝑓𝑥_𝑘,𝑃_Ω𝑥_𝑘−𝑓𝑥_𝑘−𝑣≥0, (61) that is, 𝑥_𝑘−𝑓𝑥_𝑘−𝑦_𝑘,𝑦_𝑘−𝑣≥0. (62) Note that 𝑢−𝑓(𝑣)∈𝑁_Ω(𝑣) and 𝑡_𝑘∈Ω, then 𝑣−𝑡_𝑘𝑖≥,𝑢𝑣−𝑡_𝑘𝑖(𝑣≥,𝑓)𝑣−𝑡_𝑘𝑖,𝑓(𝑣)−𝑥_𝑘𝑖−𝑓𝑥_𝑘𝑖−𝑦_𝑘𝑖,𝑦_𝑘𝑖=−𝑣𝑣−𝑡_𝑘𝑖,𝑓(𝑣)−𝑓𝑡_𝑘𝑖+𝑣−𝑡_𝑘𝑖,𝑓𝑡_𝑘𝑖−𝑣−𝑦_𝑘𝑖,𝑦_𝑘𝑖−𝑥_𝑘𝑖−𝑣−𝑦_𝑘𝑖,𝑓𝑥_𝑘𝑖≥𝑣−𝑡_𝑘𝑖,𝑓𝑡_𝑘𝑖−𝑣−𝑦_𝑘𝑖,𝑦_𝑘𝑖−𝑥_𝑘𝑖−𝑣−𝑦_𝑘𝑖,𝑓𝑥_𝑘𝑖, (63) where the last inequation follows from the monotone of 𝑓.
Since 𝑓 is continuous, by (37) we have lim_𝑖→∞𝑓𝑥_𝑘𝑖=lim_𝑖→∞𝑓𝑦_𝑘𝑖,lim_𝑖→∞𝑥_𝑘𝑖=lim_𝑖→∞𝑡_𝑘𝑖. (64) Passing onto the limit in (63), we obtain 𝑣−𝑥^∗,𝑢≥0. (65) As 𝑓 is maximal monotone, we have 𝑥^∗∈𝑓⁻¹(0), which implies that 𝑥^∗∈Ω^∗.
At last we show that such 𝑥^∗ is unique.
Let {𝑥_𝑘𝑗}^∞_𝑗=0 be another subsequence of {𝑥_𝑘}^∞_𝑘=0, such that 𝑥_𝑘𝑗⇀𝑥^∗_Δ. Then we conclude that 𝑥^∗_Δ∈𝐹(𝑆)∩Ω^∗. Suppose 𝑥^∗_Δ≠𝑥^∗; by Lemma 2 we have lim_𝑘→∞𝑥_𝑘−𝑥^∗=liminf_𝑖→∞𝑥_𝑘𝑖−𝑥^∗<liminf_𝑖→∞𝑥_𝑘𝑖−𝑥^∗_Δ=lim_𝑘→∞𝑥_𝑘−𝑥^∗_Δ=liminf_𝑗→∞𝑥_𝑘𝑗−𝑥^∗_Δ<liminf_𝑗→∞𝑥_𝑘𝑗−𝑥^∗=lim_𝑘→∞𝑥_𝑘−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 complete.

4. Further Study

In this section we propose an extension of Algorithm A, which is effective in practice. Similar to the investigation in Section 3, for the constant 𝜏>0, we define a new projected residual function as follows: 𝑟(𝑥,𝜏)∶=𝑥−𝑃_Ω𝑥−𝜏𝑓(𝑥). (67) It is clear that the new projected residual function (67) degenerates into (20) by setting 𝜏=1.

Algorithm B. Step 0. Take 𝛿∈(0,1),𝛾∈(0,1),𝜂₋₁>0,𝜃>1,𝑥₀∈Ω, and 𝑘=0.
Step 1. For the current iterative point 𝑥_𝑘∈Ω, compute 𝑦_𝑘=𝑃_Ω𝑥_𝑘−𝜏_𝑘𝑓𝑥_𝑘,𝑧_𝑘=1−𝜂_𝑘𝑥_𝑘+𝜂_𝑘𝑦_𝑘, (68) where 𝜏_𝑘=min{𝜃𝜂_𝑘−1,1},𝜂_𝑘=𝛾^𝑛𝑘𝜏_𝑘 and 𝑛_𝑘 being the smallest nonnegative integer 𝑛 satisfying 𝑓𝑥_𝑘−𝛾^𝑛𝜏_𝑘𝑟𝑥_𝑘,𝜏_𝑘,𝑟𝑥_𝑘,𝜏_𝑘≥𝛿𝜏_𝑘𝑟𝑥_𝑘,𝜏_𝑘². (69) Compute 𝑡_𝑘=𝑃_{𝐻𝑘∩Ω}𝑥_𝑘,𝑥_𝑘+1=𝛼_𝑘𝑥_𝑘+1−𝛼_𝑘𝑆𝑡_𝑘, (70) where {𝛼_𝑘}⊂(𝑎,𝑏)(𝑎,𝑏∈(0,1)) and 𝐻_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 convergence property of Algorithm B.

Lemma 10. For any 𝜏>0, one has 𝑥^∗isthesolutionofVI𝑓,Ω⟺𝑥^∗=𝑃_Ω𝑥^∗−𝜏𝑓𝑥^∗. (71)

Therefore, solving variational inequality is equivalent to finding a zero point of the projected residual function 𝑟(•,𝜏). Meanwhile we know that 𝑟(𝑥,𝜏) is a continuous function of 𝑥, as the projection mapping is nonexpansive.

Lemma 11. For any 𝑥∈𝑅^𝑛and𝜏₁≥𝜏₂>0, it holds that 𝑟𝑥,𝜏₁≥𝑟𝑥,𝜏₂. (72)

Theorem 12. Let Ω be a nonempty, closed, and convex subset of 𝐻, let 𝑓 be a monotone and 𝐿-Lipschitz continuous mapping, and 𝐹(𝑆)∩Ω^∗≠∅. Then for any sequence {𝑥_𝑘}^∞_𝑘=0,{𝑡_𝑘}^∞_𝑘=0 generated by Algorithm B, one has 𝑡_𝑘−𝑥^∗2≤𝑥_𝑘−𝑥^∗2−𝑥_𝑘−𝑡_𝑘²𝜂+2^𝑘𝑟𝑥_𝑘,𝜏_𝑘,𝑓𝑧_𝑘𝑓𝑧_𝑘²𝑧_𝑘−𝑡_𝑘,𝑓𝑧_𝑘. (73)

Proof. The proof of this theorem is similar to Theorem 7, so we omit it.

Theorem 13. Let Ω be a nonempty, closed, and convex subset of 𝐻, let 𝑓 be a monotone and 𝐿-Lipschitz continuous mapping, and 𝐹(𝑆)∩Ω^∗≠∅. Then for any sequences {𝑥_𝑘}^∞_𝑘=0,{𝑦_𝑘}^∞_𝑘=0,{𝑡_𝑘}^∞_𝑘=0 generated by Algorithm B, one has 𝑥_𝑘+1−𝑥^∗2≤𝑥_𝑘−𝑥^∗2−1−𝛼_𝑘2𝑥_𝑘−𝑡_𝑘². (74) Furthermore, lim_𝑘→∞𝑥_𝑘−𝑦_𝑘=0,lim_𝑘→∞𝑥_𝑘−𝑡_𝑘=0,lim_𝑘→∞𝑦_𝑘−𝑡_𝑘=0. (75)

Proof. The proof of this theorem is similar to the Theorem 8. The only difference is that (44) is substituted by 𝑓𝑥_𝑘−𝛾⁻¹𝜂_𝑘𝑟𝑥_𝑘,𝜏_𝑘,𝑟𝑥_𝑘,𝜏_𝑘<𝛿𝜏_𝑘𝑟𝑥_𝑘,𝜏_𝑘², (76) where (76) follows from Lemma 11 with 𝜏₁=1and𝜏₂=𝜏_𝑘.

Theorem 14. Let Ω be a nonempty, closed, and convex subset of 𝐻, let 𝑓 be a monotone and 𝐿-Lipschitz continuous mapping, and 𝐹(𝑆)∩Ω^∗≠∅. 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→∞𝑃_{𝐹(𝑆)∩Ω}^∗(𝑥_𝑘).

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 Fundamental Research Funds for the central universities (Grant: CDJXS12101103). The authors thank the anonymous reviewers for their valuable comments and suggestions, which helped to improve the paper.