Shrinking Extragradient Method for Pseudomonotone Equilibrium Problems and Quasi-Nonexpansive Mappings

Abstract

This paper presents two shrinking extragradient algorithms that can both find the solution sets of equilibrium problems for pseudomonotone bifunctions and find the sets of fixed points of quasi-nonexpansive mappings in a real Hilbert space. Under some constraint qualifications of the scalar sequences, these two new algorithms show strong convergence. Some numerical experiments are presented to demonstrate the new algorithms. Finally, the two introduced algorithms are compared with a standard, well-known algorithm.

1. Introduction

The equilibrium problem started to gain interest after the publication of a paper by Blum and Oettli [1], which discussed the problem of finding a point $x^{*}\in C$ such that

$$f(x^{*},y)\ge 0,\forall y\in C,$$

(1)

where C is a nonempty closed convex subset of a real Hilbert space H, and $f:C\times C\to (-\infty ,+\infty )$ is a bifunction. This well-known equilibrium model (1) has been used for studying a variety of mathematical models for physics, chemistry, engineering, and economics. In addition, the equilibrium problem (1) can be applied to many mathematical problems, such as optimization problems, variational inequality problems, minimax problems, Nash equilibrium problems, saddle point problems, and fixed point problems, see [1,2,3,4], and the references therein.

In order to solve the equilibrium problem (1), when f is a monotone bifunction, approximate solutions are frequently based on the proximal point method. That is, given $x_k$, at each step, the next iterate $x_{k+1}$ can be found by solving the following regularized equilibrium problem: find $x\in C$ such that

$$f(x,y)+\frac{1}{r_k}\langle y-x,x-x_k\rangle \ge 0,\forall y\in C,$$

(2)

where $\{r_k\}\subset (0,\infty )$. Note that the existence of each $x_k$ is guaranteed, on condition that the subproblem (2) is a strongly monotone problem (see [5,6]). However, if f is a pseudomonotone bifunction (a property which is weaker than a monotone) the strong monotone-ness of the problem (2) cannot be guaranteed. Therefore, the sequence $\{x_k\}$ may not be well-defined. To overcome this drawback, Tran et al. [7] proposed the following extragradient method for solving the equilibrium problem, when the considered bifunction f is pseudomonotone and Lipschitz-type continuous with positive constants $L_1$ and $L_2$:

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ y_k=argmin\{\rho f(x_k,y)+\frac{1}{2}\parallel x_k-y{\parallel }^2:y\in C\},\hfill \\ x_{k+1}=argmin\{\rho f(y_k,y)+\frac{1}{2}\parallel x_k-y{\parallel }^2:y\in C\},\hfill \end{array}\right.$$

(3)

where $0<\rho <min\left\{\frac{1}{2L_1},\frac{1}{2L_2}\right\}$. Tran et al. guaranteed that the sequence $\{x_k\}$ generated by (3) converges weakly to a solution of the equilibrium problem (1).

On the other hand, for a nonempty closed convex subset C of H and a mapping $T:C\to C$, the fixed point problem is a problem of finding a point $x\in C$ such that $Tx=x$. This fixed point problem has many important applications, such as optimization problems, variational inequality problems, minimax problems, and saddle point problems, see [8,9,10,11], and the references therein. The set of fixed points of a mapping T will be represented by $Fix\left(T\right)$.

An iteration method for finding fixed points of the mapping T was proposed by Mann [12] as follows:

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ x_{k+1}=(1-{\alpha }_k)x_k+{\alpha }_{k}T{x}_k,\hfill \end{array}\right.$$

(4)

where $\{{\alpha }_k\}\subset (0,1)$ and ${\sum }_{k=0}^{\infty }{\alpha }_k=\infty$. If T is a nonexpansive mapping and has a fixed point, then the sequence $\{x_k\}$ generated by (4) converges weakly to a fixed point of T. In addition, in 1994, Park and Jeong [13] showed that if T is a quasi-nonexpansive mapping with $I-T$ demiclosed at 0, then the sequence which is generated by the Mann iteration method converges weakly to a fixed point of T.

Furthermore, in order to obtain a strong convergence result for the Mann iteration method, Nakajo and Takahashi [14] proposed the following hybrid method:

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ y_k={\alpha }_k{x}_k+(1-{\alpha }_k)T{x}_k,\hfill \\ C_k=\{x\in C:\parallel y_k-x\parallel \le \parallel x_k-x\parallel \},\hfill \\ Q_k=\{x\in C:\langle x_0-x_k,x-x_k\rangle \le 0\},\hfill \\ x_{k+1}=P_{C_k\cap Q_k}\left(x_0\right),\hfill \end{array}\right.$$

(5)

where $\{{\alpha }_k\}\subset [0,1]$ such that ${\alpha }_k\le 1-\overline{\alpha }$ for some $\overline{\alpha }\in (0,1]$, and $P_{C_k\cap Q_k}$ is the metric projection onto $C_k\cap Q_k$. Nakajo and Takahashi proved that if T is a nonexpansive mapping, then the sequence $\{x_k\}$ generated by (5) converges strongly to $P_{Fix\left(T\right)}\left(x_0\right)$.

In addition, in 1974, Ishikawa [15] proposed the following method for finding fixed points of a Lipschitz pseudocontractive mapping T:

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ y_k=(1-{\alpha }_k)x_k+{\alpha }_{k}T{x}_k,\hfill \\ x_{k+1}=(1-{\beta }_k)x_k+{\beta }_{k}T{y}_k,\hfill \end{array}\right.$$

(6)

where $0\le {\beta }_k\le {\alpha }_k\le 1$, ${lim}_{k\to \infty }{\alpha }_k=0$ and ${\sum }_{k=0}^{\infty }{\alpha }_k{\beta }_k=\infty$. If C is a convex compact subset of H, then the sequence $\{x_k\}$ generated by (6) converges strongly to fixed points of T. It has been previously shown that the Mann iteration method is generally not applicable for finding fixed points of a Lipschitz pseudocontractive mapping in a Hilbert space. For example, see [16].

In 2008, by using Ishikawa’s iteration concept, Takahashi et al. [17] proposed the following hybrid method, called the shrinking projection method, which is different from Nakajo and Takahashi’s method [14]:

$$\left\{\begin{array}{c}{u}_0\in H,C_1=C,\hfill \\ x_1=P_{C_1}\left(u_0\right),\hfill \\ y_k={\alpha }_k{x}_k+(1-{\alpha }_k)T{x}_k,\hfill \\ z_k={\beta }_k{x}_k+(1-{\beta }_k)T{y}_k,\hfill \\ C_{k+1}=\{x\in C_k:\parallel z_k-x\parallel \le \parallel x_k-x\parallel \},\hfill \\ x_{k+1}=P_{C_{k+1}}\left(x_0\right),\hfill \end{array}\right.$$

(7)

where $\{{\alpha }_k\}\subset [\underline{\alpha },\overline{\alpha }]$ with $0<\underline{\alpha }\le \overline{\alpha }<1$, and $\{{\beta }_k\}\subset [0,1-\overline{\beta }]$ for some $\overline{\beta }\in (0,1)$. Takahashi et al. proved that if T is a nonexpansive mapping, then the sequence $\{x_k\}$ generated by (7) converges strongly to $P_{Fix\left(T\right)}\left(x_0\right)$.

In recent years, many algorithms have been proposed for finding a common element of the set of solutions of the equilibrium problem and the set of solutions of the fixed point problem. See, for instance, [8,11,18,19,20,21,22,23] and the references therein. In 2016, by using both hybrid and extragradient methods together in combination with Ishikawa’s iteration concept, Dinh and Kim [24] proposed the following iteration method for finding a common element of fixed points of a symmetric generalized hybrid mapping T and the set of solutions of the equilibrium problem, when a bifunction f is pseudomonotone and Lipschitz-type continuous with positive constants $L_1$ and $L_2$:

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ y_k=argmin\{{\rho }_{k}f(x_k,y)+\frac{1}{2}\parallel x_k-y{\parallel }^2:y\in C\},\hfill \\ z_k=argmin\{{\rho }_{k}f(y_k,y)+\frac{1}{2}\parallel x_k-y{\parallel }^2:y\in C\},\hfill \\ t_k={\alpha }_k{x}_k+(1-{\alpha }_k)T{x}_k,\hfill \\ u_k={\beta }_k{t}_k+(1-{\beta }_k)T{z}_k,\hfill \\ C_k=\{x\in H:\parallel x-u_k\parallel \le \parallel x-x_k\parallel \},\hfill \\ Q_k=\{x\in H:\langle x-x_k,x_0-x_k\rangle \le 0\},\hfill \\ x_{k+1}=P_{C_k\cap Q_k\cap C}\left(x_0\right),\hfill \end{array}\right.$$

(8)

where $\{{\rho }_k\}\subset [\underline{\rho },\overline{\rho }]$ with $0<\underline{\rho }\le \overline{\rho }<min\left\{\frac{1}{2L_1},\frac{1}{2L_2}\right\}$, $\{{\alpha }_k\}\subset [0,1]$ such that ${lim}_{k\to \infty }{\alpha }_k=1$, and $\{{\beta }_k\}\subset [0,1-\overline{\beta }]$ for some $\overline{\beta }\in (0,1)$. Dinh and Kim proved that the sequence $\{x_k\}$ generated by (8) converges strongly to $P_{EP(f,C)\cap Fix\left(T\right)}\left(x_0\right)$, where $EP(f,C)$ is the solution set of the equilibrium problem.

Now, let us consider the problem of finding a common solution of a finite family of equilibrium problems (CSEP). Let C be a nonempty closed convex subset of H and let $f_i:C\times C\to (-\infty ,+\infty )$, $i=1,\dots ,N$, be bifunctions satisfying $f_i(x,x)=0$ for each $x\in C$. The problem CSEP is to find $x^{*}\in C$ such that

$$f_i(x^{*},y)\ge 0,\forall y\in C,i=1,\dots ,N.$$

(9)

The solution set of the problem CSEP will be denoted by ${\cap }_{i=1}^{N}EP(f_i,C)$. It is worth pointing out that the problem CSEP is a generalization of many mathematical models, such as common solutions to variational inequality problems, convex feasibility problems and common fixed point problems. See [1,25,26,27] for more details. In 2016, Hieu et al. [28] considered the following problem:

$$\left\{\begin{array}{c}\mathrm{find} \mathrm{a} \mathrm{point}{x}^{*}\in C\mathrm{such} \mathrm{that}{T}_j{x}^{*}=x^{*},j=1,\dots ,M,\hfill \\ \mathrm{and}{f}_i(x^{*},y)\ge 0,\forall y\in C,i=1,\dots ,N,\hfill \end{array}\right.$$

(10)

where C is a nonempty closed convex subset of H, $T_j:C\to C$, $j=1,\dots ,M$, are mappings, and $f_i:C\times C\to (-\infty ,+\infty )$, $i=1,\dots ,N$, are bifunctions satisfying $f_i(x,x)=0$ for each $x\in C$. From now on, the solution set of problem (10) will be denoted by S. That is:

$$S:=\left({\cap }_{j=1}^{M}Fix\left(T_j\right)\right)\cap \left({\cap }_{i=1}^{N}EP(f_i,C)\right).$$

By using both hybrid and extragradient methods together in combination with Mann’s iteration concept and parallel splitting-up techniques (see [25,29]), they proposed the following algorithm for finding the solution set of problem (10), when mappings are nonexpansive, and bifunctions are pseudomonotone and Lipschitz-type continuous with positive constants $L_1$ and $L_2$:

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ y_k^i=argmin\{\rho f_i(x_k,y)+\frac{1}{2}\parallel x_k-y{\parallel }^2:y\in C\},i=1,2,\dots ,N,\hfill \\ z_k^i=argmin\{\rho f_i(y_k^i,y)+\frac{1}{2}\parallel x_k-y{\parallel }^2:y\in C\},i=1,2,\dots ,N,\hfill \\ {\overline{z}}_k=argmax\{\parallel z_k^i-x_k\parallel :i=1,2,\dots ,N\},\hfill \\ u_k^j={\alpha }_k{x}_k+(1-{\alpha }_k)T_j{\overline{z}}_k,j=1,2,\dots ,M,\hfill \\ {\overline{u}}_k=argmax\{\parallel u_k^j-x_k\parallel :j=1,2,\dots ,M\},\hfill \\ C_k=\{x\in C:\parallel x-{\overline{u}}_k\parallel \le \parallel x-x_k\parallel \},\hfill \\ Q_k=\{x\in C:\langle x-x_k,x_0-x_k\rangle \le 0\},\hfill \\ x_{k+1}=P_{C_k\cap Q_k}\left(x_0\right),\hfill \end{array}\right.$$

(11)

where $0<\rho <min\left\{\frac{1}{2L_1},\frac{1}{2L_2}\right\}$, and $\left\{{\alpha }_k\right\}\subset (0,1)$ such that $lim{sup}_{k\to \infty }{\alpha }_k<1$. Hieu et al. proved that the sequence $\left\{x_k\right\}$ generated by (PHMEM) converges strongly to $P_S\left(x_0\right)$. The algorithm (11) is called PHMEM method.

The current study will continue developing methods for finding the solution set of problem (10). Roughly speaking, some new iterative algorithms will be introduced for finding the solution set of problem (10). Some numerical examples will be considered and the introduced methods will be discussed and compared with the PHMEM algorithm.

This paper is organized as follows: In Section 2, some relevant definitions and properties will be reviewed for use in subsequent sections. Section 3 will present two shrinking extragradient algorithms and prove their convergence. Finally, in Section 4, the performance of the introduced algorithms will be compared to the performance of the PHMEM algorithm and discussed.

2. Preliminaries

This section will present some definitions and properties that will be used subsequently. First, let H be a real Hilbert space induced by the inner product $\langle ·,·\rangle$ and norm $\parallel ·\parallel$. The symbols → and ⇀ will be used here to denote the strong convergence and the weak convergence in H, respectively.

Now, recalled here are definitions of nonlinear mappings related to this work.

Definition 1 

([30,31]). Let C be a nonempty closed convex subset of H. A mapping $T:C\to C$ is said to be:

(i) 

pseudocontractive if

$${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+{\parallel (I-T)x-(I-T)y\parallel }^2,\forall x,y\in C,$$

where I denotes the identity operator on H.

(ii) 

Lipschitzian if there exists $L\ge 0$ such that

$$\parallel Tx-Ty\parallel \le L\parallel x-y\parallel ,\forall x,y\in C.$$

In particular, if $L=1$, then T is said to be nonexpansive.

(iii) 

quasi-nonexpansive if $Fix\left(T\right)$ is nonempty and

$$\parallel Tx-p\parallel \le \parallel x-p\parallel ,\forall x\in C,p\in Fix\left(T\right).$$

(iv) 

$(\alpha ,\beta ,\gamma ,\delta )$-symmetric generalized hybrid if there exists $\alpha ,\beta ,\gamma ,\delta \in (-\infty ,+\infty )$ such that

$$\begin{gathered} {\alpha \parallel Tx-Ty\parallel }^2{+\beta (\parallel x-Ty\parallel }^2{+\parallel y-Tx\parallel }^2{)+\gamma \parallel x-y\parallel }^2 \\ +\delta (\parallel x-Tx{\parallel }^2+\parallel y-Ty{\parallel }^2)\le 0,\forall x,y\in C. \end{gathered}$$

Definition 2. 

(see [32]) Let C be a nonempty closed convex subset of H and $T:C\to H$ be a mapping. The mapping T is said to be demiclosed at $y\in H$ if for any sequence $\{x_k\}\subset C$ with $x_k\rightharpoonup x^{*}\in C$ and $T{x}_k\to y$ imply $T{x}^{*}=y$.

Note that the class of pseudocontractive mappings includes the class of nonexpansive mappings. In addition, a nonexpansive mapping with at least one fixed point is a quasi-nonexpansive mapping, but the converse is not true. For example, see [33]. Moreover, if a $(\alpha ,\beta ,\gamma ,\delta )$-symmetric generalized hybrid mapping satisfies $(1)$ $\alpha +2\beta +\gamma \ge 0$, $(2)$ $\alpha +\beta >0$ and $(3)$ $\delta \ge 0$ then T is quasi-nonexpansive and $I-T$ demiclosed at 0 (see [34,35]). Moreover, $Fix\left(T\right)$ is closed and convex when T is a quasi-nonexpansive mapping (see [36]).

Next, we recall definitions and facts for considering the equilibruim problems.

Definition 3 

([1,4,37]). Let C be a nonempty closed convex subset of H and $f:C\times C\to (-\infty ,+\infty )$ be a bifunction. The bifunction f is said to be:

(i) 

strongly monotone on C if there exists a constant $\gamma >0$ such that

$$f(x,y)+f(y,x)\le {-\gamma \parallel x-y\parallel }^2,\forall x,y\in C;$$

(ii) 

monotone on C if

$$f(x,y)+f(y,x)\le 0,\forall x,y\in C;$$

(iii) 

pseudomonotone on C if

$$\forall x,y\in C,f(x,y)\ge 0\Rightarrow f(y,x)\le 0.$$

(iv) 

Lipshitz-type continuous on C with constants $L_1>0$ and $L_2>0$ if

$$f(x,y)+f(y,z)\ge f(x,z)-L_1{\parallel x-y\parallel }^2-L_2{\parallel y-z\parallel }^2,\forall x,y,z\in C.$$

Remark 1.

From Definition 3, we observe that (i) ⇒ (ii) ⇒ (iii). However, if f is pseudomonotone, f might not be monotone on C. For example, see [38].

For a nonempty closed convex subset C of H and a bifunction $f:C\times C\to (-\infty ,+\infty )$ satisfying $f(x,x)=0$ for each $x\in C$. In this paper, we are concerned with the following assumptions:

(A1)

f is weakly continuous on $C\times C$ in the sense that, if $x,y\in C$ and $\{x_k\}$, $\{y_k\}$ are two sequences in C converge weakly to x and y respectively, then $f(x_k,y_k)$ converges to $f(x,y)$;

(A2)

$f(x,·)$ is convex and subdifferentiable on C for each fixed $x\in C$;

(A3)

f is psuedomonotone on C;

(A4)

f is Lipshitz-type continuous on C with constants $L_1>0$ and $L_2>0$.

It is well-known that the solution set $EP(f,C)$ is closed and convex, when the bifunction f satisfies the assumptions $\left(A1\right)-\left(A3\right)$. See, for instance, [7,39,40].

The following facts are very important in order to obtain our main results.

Lemma 1 

([18]). Let $f:C\times C\to (-\infty ,+\infty )$ be satisfied $\left(A2\right)-\left(A4\right)$. If $EP(f,C)$ is nonempty set and $0<{\rho }_0<min\{\frac{1}{2L_1},\frac{1}{2L_2}\}$. Let $x_0\in C$. If $y_0$ and $z_0$ are defined by

$$\left\{\begin{array}{c}{y}_0=\arg \min \{{\rho }_{0}f(x_0,w)+\frac{1}{2}\parallel w-x_0{\parallel }^2:w\in C\},\hfill \\ z_0=\arg \min \{{\rho }_{0}f(y_0,w)+\frac{1}{2}\parallel w-x_0{\parallel }^2:w\in C\},\hfill \end{array}\right.$$

then,

(i) 

${\rho }_0$$[f(x_0,w)-f(x_0,y_0)]\ge \langle y_0-x_0,y_0-w\rangle$, for all $w\in C$;

(ii) 

$\parallel z_0{-q\parallel }^2\le \parallel x_0{-q\parallel }^2-(1-2{\rho }_0{L}_1)\parallel x_0-y_0{\parallel }^2-(1-2{\rho }_0{L}_2){\parallel y_0-z_0\parallel }^2$, for all $q\in EP(f,C)$.

This section will be closed by recalling the projection mapping and calculus concepts in Hilbert space.

Let C be a nonempty closed convex subset of H. For each $x\in H$, we denote the metric projection of x onto C by $P_C\left(x\right)$, that is

$$\parallel x-P_C\left(x\right)\parallel \le \parallel y-x\parallel ,\forall y\in C.$$

The following facts will also be used in this paper.

Lemma 2.

(see, for instance, [41,42]) Let C be a nonempty closed convex subset of H. Then

(i) 

$P_C\left(x\right)$ is singleton and well-defined for each $x\in H$;

(ii) 

$z=P_C\left(x\right)$ if and only if $\langle x-z,y-z\rangle \le 0$, $\forall y\in C$;

(iii) 

$\parallel P_C\left(x\right)-P_C{\left(y\right)\parallel }^2\le {\parallel x-y\parallel }^2-{\parallel P_C\left(x\right)-x+y-P_C\left(y\right)\parallel }^2$, $\forall x,y\in C$.

For a nonempty closed convex subset C of H and a convex function $g:C\to \mathbb{R}$, the subdifferential of g at $z\in C$ is defined by

$$\partial g\left(z\right)=\{w\in C:g(y)-g(z)\ge \langle w,y-z\rangle ,\forall y\in C\}.$$

The function g is said to be subdifferentiable at z if $\partial g\left(z\right)\ne \varnothing$.

3. Main Result

In this section, we propose two shrinking extragradient algorithms for finding a solution of problem (10), when each mapping $T_j$, $j=1,2,\dots ,M$, is quasi-nonexpansive with $I-T_j$ demiclosed at 0, and each bifunction $f_i$, $i=1,2,\dots ,N$, satisfies all the assumptions $\left(A1\right)-\left(A4\right)$. We start by observing that if each bifunction $f_i$, $i=1,2,\dots ,N$, is Lipshitz-type continuous on C with constants $L_1^i>0$ and $L_2^i>0$, then

$$\begin{array}{ccc}\hfill f_i(x,y)+f_i(y,z)& \ge & f_i(x,z)-L_1^i{\parallel x-y\parallel }^2-L_2^i{\parallel y-z\parallel }^2\hfill \\ & \ge & f_i(x,z)-L_1{\parallel x-y\parallel }^2-L_2{\parallel y-z\parallel }^2,\hfill \end{array}$$

where $L_1=max\{L_1^i:i=1,2,\dots ,N\}$ and $L_2=max\{L_2^i:i=1,2,\dots ,N\}$. This means the bifunctions $f_i$, $i=1,2,\dots ,N$, are Lipshitz-type continuous on C with constants $L_1>0$ and $L_2>0$. Of course, we will use this notation in this paper. Moreover, for each $N\in \mathbb{N}$ and $k\in \mathbb{N}\cup \{0\}$, we denote ${[k]}_N$ for a modulo function at k with respect to N, that is

$${[k]}_N=k\left(modN\right)+1.$$

Now, we propose a following cyclic algorithm.

CSEM Algorithm (Cyclic Shrinking Extragradient Method)

Initialization. Pick $x_0\in C=:C_0$, choose parameters $\{{\rho }_k\}$ with $0<inf{\rho }_k\le sup{\rho }_k<min\{\frac{1}{2L_1},\frac{1}{2L_2}\}$, $\{{\alpha }_k\}\subset [0,1]$ such that ${lim}_{k\to \infty }{\alpha }_k=1$, and $\{{\beta }_k\}$ with $0\le inf{\beta }_k\le sup{\beta }_k<1$.

Step 1. Solve the strongly convex program

$$y_k=argmin\{{\rho }_k{f}_{{\left[k\right]}_N}(x_k,y)+\frac{1}{2}\parallel y-x_k{\parallel }^2:y\in C\}.$$

Step 2. Solve the strongly convex program

$$z_k=argmin\{{\rho }_k{f}_{{\left[k\right]}_N}(y_k,y)+\frac{1}{2}\parallel y-x_k{\parallel }^2:y\in C\}.$$

Step 3. Compute

$$\begin{array}{c}{t}_k={\alpha }_k{x}_k+(1-{\alpha }_k)T_{{\left[k\right]}_M}{x}_k,\hfill \\ u_k={\beta }_k{t}_k+(1-{\beta }_k)T_{{\left[k\right]}_M}{z}_k.\hfill \end{array}$$

Step 4. Construct closed convex subset of C:

$$C_{k+1}=\{x\in C_k:\parallel x-u_k\parallel \le \parallel x-x_k\parallel \}.$$

Step 5. The next approximation $x_{k+1}$ is defined as the projection of $x_0$ onto $C_{k+1}$, i.e.,

$$x_{k+1}=P_{C_{k+1}}\left(x_0\right).$$

Step 6. Put $k=k+1$ and go to Step 1.

Before going to prove the strong convergence of CSEM Algorithm, we need the following lemma.

Lemma 3.

Suppose that the solution set S is nonempty. Then, the sequence $\{x_k\}$ which is generated by CSEM Algorithm is well-defined.

Proof. 

To prove the Lemma, it suffices to show that $C_k$ is a nonempty closed and convex subset of H, for each $k\in \mathbb{N}\cup \{0\}$. Firstly, we will show the non-emptiness by showing that $S\subset C_k$, for each $k\in \mathbb{N}\cup \{0\}$. Obviously, $S\subset C_0$.

Now, let $q\in S$. Then, by Lemma 1 (ii), we have

$$\parallel z_k{-q\parallel }^2\le \parallel x_k{-q\parallel }^2-(1-2{\rho }_k{L}_1)\parallel x_k-y_k{\parallel }^2-(1-2{\rho }_k{L}_2){\parallel y_k-z_k\parallel }^2,$$

for each $k\in \mathbb{N}\cup \{0\}$. This implies that

$$\parallel z_k-q\parallel \le \parallel x_k-q\parallel ,$$

(12)

for each $k\in \mathbb{N}\cup \{0\}$. On the other hand, since $q\in Fix\left(T_j\right)$, it follows from the quasi-nonexpansivity of each $T_j$ ($j\in \{1,2,\dots ,M\}$) and the definitions of $t_k$, $u_k$ that

$$\begin{array}{ccc}\hfill \parallel t_k-q\parallel & \le & {\alpha }_k\parallel x_k-q\parallel +(1-{\alpha }_k)\parallel T_{{\left[k\right]}_M}{x}_k-q\parallel \hfill \\ & \le & {\alpha }_k\parallel x_k-q\parallel +(1-{\alpha }_k)\parallel x_k-q\parallel \hfill \\ & =& \parallel x_k-q\parallel ,\hfill \end{array}$$

(13)

and

$$\begin{array}{ccc}\hfill \parallel u_k-q\parallel & \le & {\beta }_k\parallel t_k-q\parallel +(1-{\beta }_k)\parallel T_{{\left[k\right]}_M}{z}_k-q\parallel \hfill \\ & \le & {\beta }_k\parallel t_k-q\parallel +(1-{\beta }_k)\parallel z_k-q\parallel ,\hfill \end{array}$$

for each $k\in \mathbb{N}\cup \{0\}$. The relations (12) and (13) imply that

$$\begin{array}{ccc}\hfill \parallel u_k-q\parallel & \le & {\beta }_k\parallel x_k-q\parallel +(1-{\beta }_k)\parallel x_k-q\parallel \hfill \\ & =& \parallel x_k-q\parallel ,\hfill \end{array}$$

(14)

for each $k\in \mathbb{N}\cup \{0\}$. Now, suppose that $S\subset C_k$. Thus, by using (14), we see that $S\subset C_{k+1}$. So, by induction, we have $S\subset C_k$, for each $k\in \mathbb{N}\cup \{0\}$. Since S is a nonempty set, we obtain that $C_k$ is a nonempty set, for each $k\in \mathbb{N}\cup \{0\}$.

Next, we show that $C_k$ is a closed and convex subset, for each $k\in \mathbb{N}\cup \{0\}$. Note that we already have that $C_0$ is a closed and convex subset. Now, suppose that $C_k$ is a closed and convex subset, we will show that $C_{k+1}$ is likewise. To do this, let us consider a set $B_k=\{x\in H:\parallel x-u_k\parallel \le \parallel x-x_k\parallel \}$. We see that

$$B_k=\{x\in H:\langle x_k-u_k,x\rangle \le {\displaystyle \frac{1}{2}}(\parallel x_k{\parallel }^2-\parallel u_k{\parallel }^2\left)\right\}.$$

This means that $B_k$ is a halfspace and $C_{k+1}=C_k\cap B_k$. Thus, $C_{k+1}$ is a closed and convex subset. Thus, by induction, we can conclude that $C_k$ is a closed and convex subset, for each $k\in \mathbb{N}\cup \{0\}$. Consequently, we can guarantee that $\left\{x_k\right\}$ is well-defined.  □

Theorem 1.

Suppose that the solution set S is nonempty. Then, the sequence $\left\{x_k\right\}$ which is generated by CSEM Algorithm converges strongly to $P_S\left(x_0\right)$.

Proof. 

Let $q\in S$. By the definition of $x_{k+1}$, we observe that $x_{k+1}\in C_{k+1}\subset C_k$, for each $k\in \mathbb{N}\cup \{0\}$. Since $x_k=P_{C_k}\left(x_0\right)$ and $x_{k+1}\in C_k$, we have

$$\parallel x_k-x_0\parallel \le \parallel x_{k+1}-x_0\parallel ,$$

for each $k\in \mathbb{N}\cup \{0\}$. This means that $\{\parallel x_k-x_0\parallel \}$ is a nondecreasing sequence. Similarly, for each $q\in S\subset C_{k+1}$, we obtain that

$$\parallel x_{k+1}-x_0\parallel \le \parallel q-x_0\parallel ,$$

for each $k\in \mathbb{N}\cup \{0\}$. By the above inequalities, we get

$$\parallel x_k-x_0\parallel \le \parallel q-x_0\parallel ,$$

(15)

for each $k\in \mathbb{N}\cup \{0\}$. So $\{\parallel x_k-x_0\parallel \}$ is a bounded sequence. Consequently, we can conclude that $\{\parallel x_k-x_0\parallel \}$ is a convergent sequence. Moreover, we see that $\left\{x_k\right\}$ is bounded. Thus, in view of (13) and (14), we get that $\left\{t_k\right\}$ and $\left\{u_k\right\}$ are also bounded. Suppose $k,j\in \mathbb{N}\cup \{0\}$ such that $k>j$. It follows that $x_k\in C_k\subset C_j$. Then, by Lemma 2 (iii), we have

$$\parallel P_{C_j}\left(x_k\right)-P_{C_j}\left(x_0\right){\parallel }^2\le \parallel x_0-x_k{\parallel }^2-{\parallel P_{C_j}\left(x_k\right)-x_k+x_0-P_{C_j}\left(x_0\right)\parallel }^2.$$

Consequently,

$$\parallel x_k-x_j{\parallel }^2\le \parallel x_0-x_k{\parallel }^2-{\parallel x_j-x_0\parallel }^2.$$

Thus, by using the existence of ${lim}_{k\to \infty }\parallel x_k-x_0\parallel$, we get

$$\underset{k,j\to \infty }{lim}\parallel x_k-x_j\parallel =0.$$

That is $\left\{x_k\right\}$ is a Cauchy sequence in C. Since C is closed, there exists $p\in C$ such that

$$\underset{k\to \infty }{lim}{x}_k=p.$$

(16)

By the definition of $C_{k+1}$ and $x_{k+1}\in C_k$, we see that

$$\parallel x_{k+1}-u_k\parallel \le \parallel x_{k+1}-x_k\parallel ,$$

for each $k\in \mathbb{N}\cup \{0\}$. It follows that

$$\begin{array}{ccc}\hfill \parallel u_k-x_k\parallel & \le & \parallel u_k-x_{k+1}\parallel +\parallel x_{k+1}-x_k\parallel \hfill \\ & \le & \parallel x_{k+1}-x_k\parallel +\parallel x_{k+1}-x_k\parallel \hfill \\ & =& 2\parallel x_{k+1}-x_k\parallel ,\hfill \end{array}$$

(17)

for each $k\in \mathbb{N}\cup \{0\}$. Since $x_k\to p$ and $x_{k+1}\to p$, as $k\to \infty$, we obtain that

$$\underset{k\to \infty }{lim}\parallel x_{k+1}-x_k\parallel =0.$$

This together with (17) imply that

$$\underset{k\to \infty }{lim}\parallel u_k-x_k\parallel =0.$$

(18)

Since ${lim}_{k\to \infty }{\alpha }_k=1$ and the quasi-nonexpansivity of each $T_j$ ($j\in \{1,2,\dots ,M\}$), it follows that

$$\begin{array}{ccc}\hfill \underset{k\to \infty }{lim}\parallel t_k-x_k\parallel & =& \underset{k\to \infty }{lim}\parallel {\alpha }_k{x}_k+(1-{\alpha }_k)T_{{\left[k\right]}_M}{x}_k-x_k\parallel \hfill \\ & =& \underset{k\to \infty }{lim}(1-{\alpha }_k)\parallel x_k-T_{{\left[k\right]}_M}{x}_k\parallel \hfill \\ & =& 0.\hfill \end{array}$$

(19)

Consider,

$$\begin{array}{ccc}\hfill \parallel u_k{-q\parallel }^2& =& \parallel {\beta }_k(t_k-q)+(1-{\beta }_k)(T_{{\left[k\right]}_M}{z}_k-q){\parallel }^2\hfill \\ & =& {\beta }_k\parallel t_k{-q\parallel }^2+(1-{\beta }_k)\parallel T_{{\left[k\right]}_M}{z}_k{-q\parallel }^2-{\beta }_k(1-{\beta }_k){\parallel t_k-T_{{\left[k\right]}_M}{z}_k\parallel }^2\hfill \\ & \le & {\beta }_k\parallel t_k{-q\parallel }^2+(1-{\beta }_k){\parallel T_{{\left[k\right]}_M}{z}_k-q\parallel }^2,\hfill \end{array}$$

for each $k\in \mathbb{N}\cup \{0\}$. By using (13) and the quasi-nonexpansivity of each $T_j$ ($j\in \{1,2,\dots ,M\}$), we obtain

$$\parallel u_k{-q\parallel }^2\le {\beta }_k\parallel x_k{-q\parallel }^2+(1-{\beta }_k){\parallel z_k-q\parallel }^2,$$

for each $k\in \mathbb{N}\cup \{0\}$. Then, by Lemma 1 (ii), we have

$$\begin{array}{ccc}\hfill \parallel u_k{-q\parallel }^2& \le & {\beta }_k\parallel x_k{-q\parallel }^2+(1-{\beta }_k)[\parallel x_k{-q\parallel }^2-(1-2{\rho }_k{L}_1)\parallel x_k-y_k{\parallel }^2-(1-2{\rho }_k{L}_2)\parallel y_k-z_k{\parallel }^2]\hfill \\ & \le & \parallel x_k{-q\parallel }^2-(1-{\beta }_k)[(1-2{\rho }_k{L}_1)\parallel x_k-y_k{\parallel }^2+(1-2{\rho }_k{L}_2)\parallel y_k-z_k{\parallel }^2],\hfill \end{array}$$

for each $k\in \mathbb{N}\cup \{0\}$. It follows that

$$\begin{array}{c}\hfill (1-{\beta }_k)[(1-2{\rho }_k{L}_1)\parallel x_k-y_k{\parallel }^2+(1-2{\rho }_k{L}_2)\parallel y_k-z_k{\parallel }^2]\le \parallel x_k-u_k\parallel (\parallel x_k-q\parallel +\parallel u_k-q\parallel ),\end{array}$$

(20)

for each $k\in \mathbb{N}\cup \{0\}$. Thus, by using (18) and the choices of $\left\{{\beta }_k\right\}$, $\left\{{\rho }_k\right\}$, we have

$$\underset{k\to \infty }{lim}\parallel x_k-y_k\parallel =0,$$

(21)

and

$$\underset{k\to \infty }{lim}\parallel y_k-z_k\parallel =0.$$

(22)

These imply that

$$\underset{k\to \infty }{lim}\parallel x_k-z_k\parallel =0.$$

(23)

Then, by ${lim}_{k\to \infty }{x}_k=p$, we also have

$$\underset{k\to \infty }{lim}{y}_k=p,$$

(24)

and

$$\underset{k\to \infty }{lim}{z}_k=p.$$

(25)

Next, we claim that $p\in S$. From the definition of $u_k$, we see that

$$\begin{array}{ccc}\hfill (1-{\beta }_k)\parallel T_{{[k]}_M}{z}_k-z_k\parallel & =& \parallel u_k-z_k-{\beta }_k(t_k-z_k)\parallel \hfill \\ & \le & \parallel u_k-z_k\parallel +{\beta }_k\parallel t_k-z_k\parallel \hfill \\ & \le & \parallel u_k-x_k\parallel +{\beta }_k\parallel t_k-x_k\parallel +(1+{\beta }_k)\parallel x_k-z_k\parallel ,\hfill \end{array}$$

for each $k\in \mathbb{N}\cup \{0\}$. Then, by using (18), (19) and (23), we have

$$\underset{k\to \infty }{lim}\parallel T_{{[k]}_M}{z}_k-z_k\parallel =0.$$

(26)

Furthermore, for each fixed $j\in \{1,2,\dots ,M\}$, we observe that

$${[(j-1)+kM]}_M=j,$$

for each $k\in \mathbb{N}\cup \{0\}$. Thus, it follows from (26) that

$$\begin{array}{ccc}\hfill 0& =& \underset{k\to \infty }{lim}\parallel T_{{[(j-1)+kM]}_M}{z}_{(j-1)+kM}-z_{(j-1)+kM}\parallel \hfill \\ & =& \underset{k\to \infty }{lim}\parallel T_j{z}_{(j-1)+kM}-z_{(j-1)+kM}\parallel ,\hfill \end{array}$$

(27)

for each $j\in \{1,2,\dots ,M\}$. Since $z_k\to p$, as $k\to \infty$, then for each $j\in \{1,2,\dots ,M\}$, we get $z_{(j-1)+kM}\to p$, as $k\to \infty$. Combining with (27), by the demiclosedness at 0 of $I-T_j$, implies that

$$T_{j}p=p,$$

for each $j=1,2,\dots ,M$.

Similarly, for each fixed $i\in \{1,2,\dots ,N\}$, we note that

$${[(i-1)+kN]}_N=i,$$

for each $k\in \mathbb{N}\cup \{0\}$. Since $x_k\to p$ and $y_k\to p$, as $k\to \infty$, then for each $i\in \{1,2,\dots ,N\}$, we have $x_{(i-1)+kN}\to p$ and $y_{(i-1)+kN}\to p$, as $k\to \infty$. By Lemma 1 (i), for each $i\in \{1,2,\dots ,N\}$, we obtain

$$\begin{array}{c}{\rho }_{(i-1)+kN}[f_{{[(i-1)+kN]}_N}(x_{(i-1)+kN},y)-f_{{[(i-1)+kN]}_N}(x_{(i-1)+kN},y_{(i-1)+kN})]\hfill \\ \ge \langle y_{(i-1)+kN}-x_{(i-1)+kN},y_{(i-1)+kN}-y\rangle ,\forall y\in C.\hfill \end{array}$$

It follows that, for each $i\in \{1,2,\dots ,N\}$, we have

$$\begin{array}{c}{f}_{{[(i-1)+kN]}_N}(x_{(i-1)+kN},y)-f_{{[(i-1)+kN]}_N}(x_{(i-1)+kN},y_{(i-1)+kN})\hfill \\ \ge -{\displaystyle \frac{1}{{\rho }_{(i-1)+kN}}}\parallel y_{(i-1)+kN}-x_{(i-1)+kN}\parallel \parallel y_{(i-1)+kN}-y\parallel ,\forall y\in C.\hfill \end{array}$$

By using (21) and weak continuity of each $f_i$ ($i\in \{1,2,\dots ,N\}$), we get that

$$f_i(p,y)\ge 0,\forall y\in C,$$

for each $i=1,2,\dots ,N$. Then, we had shown that $p\in S$.

Finally, we will show that $p=P_S\left(x_0\right)$. In fact, since $P_S\left(x_0\right)\in S$, it follows from (15) that

$$\parallel x_k-x_0\parallel \le \parallel P_S\left(x_0\right)-x_0\parallel ,$$

for each $k\in \mathbb{N}\cup \{0\}$. Then, by using the continuity of norm and ${lim}_{k\to \infty }{x}_k=p$, we see that

$$\parallel p-x_0\parallel =\underset{k\to \infty }{lim}\parallel x_k-x_0\parallel \le \parallel P_S\left(x_0\right)-x_0\parallel .$$

Thus, by the definition of $P_S\left(x_0\right)$ and $p\in S$, we obtain that $p=P_S\left(x_0\right)$. This completes the proof.  □

Next, by replacing cyclic method by parallel method, we propose the following algorithm.

PSEM Algorithm (Parallel Shrinking Extragradient Method)

Initialization. Pick $x_0\in C=:C_0$, choose parameters $\left\{{\rho }_k^i\right\}$ with $0<inf{\rho }_k^i\le sup{\rho }_k^i<min\{\frac{1}{2L_1},\frac{1}{2L_2}\},i=1,2,\dots ,N$, $\left\{{\alpha }_k\right\}\subset [0,1]$ such that ${lim}_{k\to \infty }{\alpha }_k=1$, and $\left\{{\beta }_k\right\}$ with $0\le inf{\beta }_k\le sup{\beta }_k<1$.

Step 1. Solve N strongly convex programs

$$\begin{array}{c}{y}_k^i=argmin\{{\rho }_k^i{f}_i(x_k,y)+\frac{1}{2}\parallel y-x_k{\parallel }^2:y\in C\},i=1,2,\dots ,N.\hfill \end{array}$$

Step 2. Solve N strongly convex programs

$$\begin{array}{c}{z}_k^i=argmin\{{\rho }_k^i{f}_i(y_k^i,y)+\frac{1}{2}\parallel y-x_k{\parallel }^2:y\in C\},i=1,2,\dots ,N.\hfill \end{array}$$

Step 3. Find the farthest element from $x_k$ among $z_k^i$, $i=1,2,\dots ,N$, i.e.,

$$\begin{array}{c}\hfill {\overline{z}}_k=argmax\{\parallel z_k^i-x_k\parallel :i=1,2,\dots ,N\}.\end{array}$$

Step 4. Compute

$$\begin{array}{c}{t}_k^j={\alpha }_k{x}_k+(1-{\alpha }_k)T_j{x}_k,j=1,2,\dots ,M,\hfill \\ u_k^j={\beta }_k{t}_k^j+(1-{\beta }_k)T_j{\overline{z}}_k,j=1,2,\dots ,M.\hfill \end{array}$$

Step 5. Find the farthest element from $x_k$ among $u_k^j$, $j=1,2,\dots ,M$, i.e.,

$$\begin{array}{c}\hfill {\overline{u}}_k=argmax\{\parallel u_k^j-x_k\parallel :j=1,2,\dots ,M\}.\end{array}$$

Step 6. Construct closed convex subset of C:

$$C_{k+1}=\{x\in C_k:\parallel x-{\overline{u}}_k\parallel \le \parallel x-x_k\parallel \}.$$

Step 7. The next approximation $x_{k+1}$ is defined as the projection of $x_0$ onto $C_{k+1}$, i.e.,

$$x_{k+1}=P_{C_{k+1}}\left(x_0\right)$$

.

Step 8. Put $k=k+1$ and go to Step 1.

Theorem 2.

Suppose that the solution set S is nonempty. Then, the sequence $\left\{x_k\right\}$ which is generated by PSEM Algorithm converges strongly to $P_S\left(x_0\right)$.

Proof. 

Let $q\in S$. By the definition of ${\overline{z}}_k$, we suppose that $i_k\in \{1,2,\dots ,N\}$ such that $z_k^{i_k}={\overline{z}}_k=argmax\{\parallel z_k^i-x_k\parallel :i=1,2,\dots ,N\}$. Then, by Lemma 1 (ii), we have

$$\parallel {\overline{z}}_k{-q\parallel }^2\le \parallel x_k{-q\parallel }^2-(1-2{\rho }_k^{i_k}{L}_1)\parallel x_k-y_k^{i_k}{\parallel }^2-(1-2{\rho }_k^{i_k}{L}_2){\parallel y_k^{i_k}-{\overline{z}}_k\parallel }^2,$$

for each $k\in \mathbb{N}\cup \{0\}$. This implies that

$$\begin{array}{c}\hfill \parallel {\overline{z}}_k-q\parallel \le \parallel x_k-q\parallel ,\end{array}$$

(28)

for each $k\in \mathbb{N}\cup \{0\}$. On the other hand, by the definition of $t_k^j$ and the quasi-nonexpansivity of each $T_j$ ($j\in \{1,2,\dots ,M\}$), we have

$$\begin{array}{ccc}\hfill \parallel t_k^j-q\parallel & \le & {\alpha }_k\parallel x_k-q\parallel +(1-{\alpha }_k)\parallel T_j{x}_k-q\parallel \hfill \\ & \le & {\alpha }_k\parallel x_k-q\parallel +(1-{\alpha }_k)\parallel x_k-q\parallel \hfill \\ & =& \parallel x_k-q\parallel ,\hfill \end{array}$$

(29)

for each $k\in \mathbb{N}\cup \{0\}$. Additionally, by the definition of ${\overline{u}}_k$, we suppose that $j_k\in \{1,2,\dots ,M\}$ such that $u_k^{j_k}={\overline{u}}_k=argmax\{\parallel u_k^j-x_k\parallel :j=1,2,\dots ,M\}$. It follows from the quasi-nonexpansivity of each $T_j$ ($j\in \{1,2,\dots ,M\}$) that

$$\begin{array}{ccc}\hfill \parallel {\overline{u}}_k-q\parallel & \le & {\beta }_k\parallel t_k^{j_k}-q\parallel +(1-{\beta }_k)\parallel T_{j_k}{\overline{z}}_k-q\parallel \hfill \\ & \le & {\beta }_k\parallel t_k^{j_k}-q\parallel +(1-{\beta }_k)\parallel {\overline{z}}_k-q\parallel ,\hfill \end{array}$$

for each $k\in \mathbb{N}\cup \{0\}$. The relations (28) and (29) imply that

$$\begin{array}{ccc}\hfill \parallel {\overline{u}}_k-q\parallel & \le & {\beta }_k\parallel x_k-q\parallel +(1-{\beta }_k)\parallel x_k-q\parallel \hfill \\ & =& \parallel x_k-q\parallel ,\hfill \end{array}$$

(30)

for each $k\in \mathbb{N}\cup \{0\}$. Following the proof of Lemma 3 and Theorem 1, we can show that $C_k$ is a closed convex subset of H and $S\subset C_k$, for each $k\in \mathbb{N}\cup \{0\}$. Moreover, we can check that the sequence $\left\{x_k\right\}$ is a convergent sequence, say

$$\underset{k\to \infty }{lim}{x}_k=p,$$

(31)

for some $p\in C$.

By the definition of $C_{k+1}$ and $x_{k+1}\in C_k$, we see that

$$\parallel x_{k+1}-{\overline{u}}_k\parallel \le \parallel x_{k+1}-x_k\parallel ,$$

for each $k\in \mathbb{N}\cup \{0\}$. It follows that

$$\begin{array}{ccc}\hfill \parallel {\overline{u}}_k-x_k\parallel & \le & \parallel {\overline{u}}_k-x_{k+1}\parallel +\parallel x_{k+1}-x_k\parallel \hfill \\ & \le & \parallel x_{k+1}-x_k\parallel +\parallel x_{k+1}-x_k\parallel \hfill \\ & =& 2\parallel x_{k+1}-x_k\parallel ,\hfill \end{array}$$

(32)

for each $k\in \mathbb{N}\cup \{0\}$. Since $x_k\to p$ and $x_{k+1}\to p$, as $k\to \infty$, we obtain that

$$\underset{k\to \infty }{lim}\parallel x_{k+1}-x_k\parallel =0.$$

This together with (32) implies that

$$\underset{k\to \infty }{lim}\parallel {\overline{u}}_k-x_k\parallel =0.$$

Then, by the definition of ${\overline{u}}_k$, we have

$$\underset{k\to \infty }{lim}\parallel u_k^j-x_k\parallel =0,$$

(33)

for each $j=1,2,\dots ,M$. Since ${lim}_{k\to \infty }{\alpha }_k=1$ and the quasi-nonexpansivity of each $T_j$ ($j\in \{1,2,\dots ,M\}$), it follows that

$$\begin{array}{ccc}\hfill \underset{k\to \infty }{lim}\parallel t_k^j-x_k\parallel & =& \underset{k\to \infty }{lim}\parallel {\alpha }_k{x}_k+(1-{\alpha }_k)T_j{x}_k-x_k\parallel \hfill \\ & =& \underset{k\to \infty }{lim}(1-{\alpha }_k)\parallel x_k-T_j{x}_k\parallel \hfill \\ & =& 0,\hfill \end{array}$$

(34)

for each $j=1,2,\dots ,M$. Beside, by the definition of $u_k^j$, for each $j=1,2,\dots ,M$, we see that

$$\begin{array}{ccc}\hfill \parallel u_k^j{-q\parallel }^2& =& \parallel {\beta }_k(t_k^j-q)+(1-{\beta }_k)(T_j{\overline{z}}_k-q){\parallel }^2\hfill \\ & =& {\beta }_k\parallel t_k^j{-q\parallel }^2+(1-{\beta }_k)\parallel T_j{\overline{z}}_k{-q\parallel }^2-{\beta }_k(1-{\beta }_k){\parallel t_k^j-T_j{\overline{z}}_k\parallel }^2\hfill \\ & \le & {\beta }_k\parallel t_k^j{-q\parallel }^2+(1-{\beta }_k){\parallel T_j{\overline{z}}_k-q\parallel }^2,\hfill \end{array}$$

for each $k\in \mathbb{N}\cup \{0\}$. Thus, by using (29) and the quasi-nonexpansivity of each $T_j$ ($j\in \{1,2,\dots ,M\}$), we have

$$\parallel u_k^j{-q\parallel }^2\le {\beta }_k\parallel x_k{-q\parallel }^2+(1-{\beta }_k){\parallel {\overline{z}}_k-q\parallel }^2,$$

for $k\in \mathbb{N}\cup \{0\}$. So, by Lemma 1 (ii), for each $j=1,2,\dots ,M$, we get that

$$\begin{array}{ccc}\hfill \parallel u_k^j{-q\parallel }^2& \le & {\beta }_k\parallel x_k{-q\parallel }^2+(1-{\beta }_k)[\parallel x_k{-q\parallel }^2-(1-2{\rho }_k^{i_k}{L}_1)\parallel x_k-y_k^{i_k}{\parallel }^2-(1-2{\rho }_k^{i_k}{L}_2)\parallel y_k^{i_k}-{\overline{z}}_k{\parallel }^2]\hfill \\ & =& \parallel x_k{-q\parallel }^2-(1-{\beta }_k)[(1-2{\rho }_k^{i_k}{L}_1)\parallel x_k-y_k^{i_k}{\parallel }^2+(1-2{\rho }_k^{i_k}{L}_2)\parallel y_k^{i_k}-{\overline{z}}_k{\parallel }^2],\hfill \end{array}$$

for each $k\in \mathbb{N}\cup \{0\}$. It follows that, for each $j=1,2,\dots ,M$, we have

$$\begin{array}{cc}& (1-{\beta }_k)[(1-2{\rho }_k^{i_k}{L}_1)\parallel x_k-y_k^{i_k}{\parallel }^2+(1-2{\rho }_k^{i_k}{L}_2)\parallel y_k^{i_k}-{\overline{z}}_k{\parallel }^2]\hfill \\ \le & \parallel x_k{-q\parallel }^2-{\parallel u_k^j-q\parallel }^2\hfill \\ =& \parallel x_k-u_k^j\parallel (\parallel x_k-q\parallel +\parallel u_k^j-q\parallel ),\hfill \end{array}$$

(35)

for each $k\in \mathbb{N}\cup \{0\}$. Thus, by using (33) and the choices of $\left\{{\beta }_k\right\}$, $\left\{{\rho }_k^i\right\}$, we see that

$$\underset{k\to \infty }{lim}\parallel x_k-y_k^{i_k}\parallel =0,$$

(36)

and

$$\underset{k\to \infty }{lim}\parallel y_k^{i_k}-{\overline{z}}_k\parallel =0.$$

(37)

These imply that

$$\underset{k\to \infty }{lim}\parallel x_k-{\overline{z}}_k\parallel =0.$$

(38)

Then, by the definition of ${\overline{z}}_k$, we have

$$\underset{k\to \infty }{lim}\parallel x_k-z_k^i\parallel =0,$$

(39)

for each $i=1,2,\dots ,N$. Moreover, by Lemma 1 (ii), for each $i=1,2,\dots ,N$, we get that

$$\begin{array}{c}\hfill \parallel z_k^i{-q\parallel }^2\le \parallel x_k{-q\parallel }^2-(1-2{\rho }_k^i{L}_1)\parallel x_k-y_k^i{\parallel }^2-(1-2{\rho }_k^i{L}_2){\parallel y_k^i-z_k^i\parallel }^2,\end{array}$$

for each $k\in \mathbb{N}\cup \{0\}$. It follows that, for each $i=1,2,\dots ,N$, we have

$$\begin{array}{ccc}\hfill (1-2{\rho }_k^i{L}_1)\parallel x_k-y_k^i{\parallel }^2+(1-2{\rho }_k^i{L}_2){\parallel y_k^i-z_k^i\parallel }^2& \le & \parallel x_k{-q\parallel }^2-{\parallel z_k^i-q\parallel }^2\hfill \\ & =& \parallel x_k-z_k^i\parallel (\parallel x_k-q\parallel +\parallel z_k^i-q\parallel ),\hfill \end{array}$$

for each $k\in \mathbb{N}\cup \{0\}$. Combining with (39) implies that

$$\underset{k\to \infty }{lim}\parallel x_k-y_k^i\parallel =0,$$

(40)

and

$$\underset{k\to \infty }{lim}\parallel y_k^i-z_k^i\parallel =0,$$

(41)

for each $i=1,2,\dots ,N$. Thus, by using (38), (40) and ${lim}_{k\to \infty }{x}_k=p$, we have

$$\underset{k\to \infty }{lim}{\overline{z}}_k=p,$$

(42)

and

$$\underset{k\to \infty }{lim}{y}_k^i=p,$$

(43)

for each $i=1,2,\dots ,N$.

Next, we claim that $p\in S$. From the definition of $u_k^j$, for each $j=1,2,\dots ,M$, we see that

$$\begin{array}{ccc}\hfill (1-{\beta }_k)\parallel T_j{\overline{z}}_k-{\overline{z}}_k\parallel & =& \parallel u_k^j-{\overline{z}}_k-{\beta }_k(t_k^j-{\overline{z}}_k)\parallel \hfill \\ & \le & \parallel u_k^j-{\overline{z}}_k\parallel +{\beta }_k\parallel t_k^j-{\overline{z}}_k\parallel \hfill \\ & \le & \parallel u_k^j-x_k\parallel +{\beta }_k\parallel t_k^j-x_k\parallel +(1+{\beta }_k)\parallel x_k-{\overline{z}}_k\parallel ,\hfill \end{array}$$

for each $k\in \mathbb{N}\cup \{0\}$. Thus, in view of (33), (34), and (38), we get that

$$\underset{k\to \infty }{lim}\parallel T_j{\overline{z}}_k-{\overline{z}}_k\parallel =0,$$

(44)

for each $j=1,2,\dots ,M$. Combining with (42), by the demiclosedness at 0 of $I-T_j$, implies that

$$T_{j}p=p,$$

for each $j=1,2,\dots ,M$.

On the other hand, by Lemma 1 (i), for each $i=1,2,\dots ,N$, we see that

$${\rho }_k^i[f_i(x_k,y)-f_i(x_k,y_k^i)]\ge \langle y_k^i-x_k,y_k^i-y\rangle ,\forall y\in C.$$

It follows that, for each $i=1,2,\dots ,N$, we get

$$f_i(x_k,y)-f_i(x_k,y_k^i)\ge -{\displaystyle \frac{1}{{\rho }_k^i}}\parallel y_k^i-x_k\parallel \parallel y_k^i-y\parallel ,\forall y\in C.$$

By using (31), (40), (43) and weak continuity of each $f_i$ ($i\in \{1,2,\dots ,N\}$), we have

$$f_i(p,y)\ge 0,\forall y\in C,$$

for each $i=1,2,\dots ,N$. Thus, we can conclude that $p\in S$. The rest of the proof is similar to the arguments in the proof of Theorem 1, and it leads to the conclusion that the sequence $\left\{x_k\right\}$ converges strongly to $P_S\left(x_0\right)$.  □

Remark 2.

We note that for the PSEM algorithm we solve $y_k^i$, $z_k^i$, $i=1,2,\dots ,N$, by using N bifunctions and compute $t_k^j$, $u_k^j$, $j=1,2,\dots ,M$, by using M mappings. The farthest elements from $x_k$ among all $z_k^i$ and $u_k^j$ are chosen for the next step calculation. However, we solve only $y_k$, $z_k$, by using a bifunction and compute only $t_k$, $u_k$, by using a mapping for the CSEM algorithm. After that, we construct closed convex subset $C_{k+1}$, and the approximation $x_{k+1}$ is the projection of $x_0$ onto $C_{k+1}$ for both algorithms. We claim that the numbers of iterations of the PSEM algorithm should be less than the CSEM algorithm. However, the computational times of the CSEM algorithm should be less than the PSEM algorithm for sufficiently large $N,M$. On the other hand, for the PHMEM algorithm they solved $y_k^i$, $z_k^i$, $i=1,2,\dots ,N$, by using N bifunctions, and computed $u_k^j$, $j=1,2,\dots ,M$, by using M mappings. The farthest elements from $x_k$ among all $z_k^i$ and $u_k^j$ are chosen similar to the PSEM algorithm. However, they constructed two closed convex subsets $C_k$, $Q_k$, and the approximation $x_{k+1}$ is the projection of $x_0$ onto $C_k\cap Q_k$, which is difficult to compute. We will focus on these observations in the next section.

4. A Numerical Experiment

This section will compare the two introduced algorithms, CSEM and PSEM, with the PHMEM algorithm, which was presented in [28]. The following setting is taken from Hieu et al. [28]. Let $H=\mathbb{R}$ be a Hilbert space with the standard inner product $\langle x,y\rangle =xy$ and the norm $\parallel x\parallel =\left|x\right|$, for each $x,y\in H$. To be considered here are the nonexpansive self-mappings $T_j$, $j=1,2,\dots ,M$, and the bifunctions $f_i$, $i=1,2,\dots ,N$, which are given on $C=[0,1]$ by

$$T_j\left(x\right)={\displaystyle \frac{x^j{sin}^{j-1}\left(x\right)}{2j-1}},j=1,2,\dots ,M,$$

and

$$f_i(x,y)=B_i\left(x\right)(y-x),i=1,2,\dots ,N,$$

where $B_i\left(x\right)=0$ if $0\le x\le {\xi }_i$, and $B_i\left(x\right)=e^{x-{\xi }_i}+sin(x-{\xi }_i)-1$ if ${\xi }_i<x\le 1$. Moreover, $0<{\xi }_1<{\xi }_2<\dots <{\xi }_N<1$. Then, the bifunctions $f_i$, $i=1,2,\dots ,N$, satisfy conditions $\left(A1\right)-\left(A4\right)$ (see [28]). Indeed, the bifunctions $f_i$, $i=1,2,\dots ,N$, are Lipshitz-type continuous with constants $L_1=L_2=2$. Note that the solution set S is nonempty because $0\in S$.

The following numerical experiment is considered with these parameters: ${\rho }_k=\frac{1}{5}$, ${\xi }_{{\left[k\right]}_N}=\frac{{\left[k\right]}_N}{N+1}$ for the CSEM algorithm; ${\rho }_k^i=\frac{1}{5}$, ${\xi }_i=\frac{i}{N+1}$, $i=1,2,\dots ,N$ for the PSEM algorithm, when $N=1000$ and $M=2000$. The following six cases of the parameters ${\alpha }_k$ and ${\beta }_k$ are considered:

Case 1. ${\alpha }_k=1-{\displaystyle \frac{1}{k+2}}$, ${\beta }_k={\displaystyle \frac{1}{k+2}}$.

Case 2. ${\alpha }_k=1-{\displaystyle \frac{1}{k+2}}$, ${\beta }_k=0.5+{\displaystyle \frac{1}{k+3}}$.

Case 3. ${\alpha }_k=1-{\displaystyle \frac{1}{k+2}}$, ${\beta }_k=0.99-{\displaystyle \frac{1}{k+2}}$.

Case 4. ${\alpha }_k=1$, ${\beta }_k={\displaystyle \frac{1}{k+2}}$.

Case 5. ${\alpha }_k=1$, ${\beta }_k=0.5+{\displaystyle \frac{1}{k+3}}$.

Case 6. ${\alpha }_k=1$, ${\beta }_k=0.99-{\displaystyle \frac{1}{k+2}}$.

The experiment was written in Matlab R2015b and performed on a PC desktop with Intel(R) Core(TM) i3-3240 CPU @ 3.40GHz 3.40GHz and RAM 4.00 GB. The function $fmincon$ in Matlab Optimization Toolbox was used to solve vectors $y_k$, $z_k$ for the CSEM algorithm; $y_k^i$, $z_k^i$, $i=1,2,\dots ,N$, for the PSEM algorithm. The set $C_{k+1}$ was computed by using the function $solve$ in Matlab Symbolic Math Toolbox. One can see that the set $C_{k+1}$ is the interval $[a,b]$, where $a,b\in [0,1]$, $a\le b$. Consequently, the metric projection of a point $x_0$ onto the set $C_{k+1}$ was computed by using this form

$$P_{C_{k+1}}\left(x_0\right)=max\{min\{x_0,b\},a\},$$

see [41]. The CSEM and PSEM algorithms were tested along with the PHMEM algorithm by using the stopping criteria $|x_{k+1}-x_k|<{10}^{-4}$ and the results below were presented as averages calculated from four starting points: $x_0$ at $0.01$, $0.25$, $0.75$ and 1.

Table 1. Numerical results for six different cases of parameters ${\alpha }_k$ and ${\beta }_k$.

	Average Times (sec)			Average Iterations
Cases	CSEM	PSEM	PHMEM	CSEM	PSEM	PHMEM
1	4.905197	165.099794	173.347257	14.25	13.75	14.25
2	7.326055	287.918141	345.025914	25.25	24.25	28.25
3	20.371064	834.001035	2004.693844	91.25	74.25	177
4	5.079676	173.091716	173.347257	14.75	14.25	14.25
5	8.016109	342.870819	345.025914	28.75	28.25	28.25
6	42.035240	1986.147273	2004.693844	200	177	177

shows that the parameter ${\beta }_k={\displaystyle \frac{1}{k+2}}$ yields faster computational times and fewer computational iterations than other cases. Compare cases 1–3 with each other and cases 4–6 with each other. Meanwhile, the parameter ${\alpha }_k=1$, in which the Ishikawa iteration reduces to the Mann iteration, yields slower computational times and more computational iterations than the other case. Compare cases 1 with 4, 2 with 5, and 3 with 6. Moreover, the computational times of the CSEM algorithm are faster than other algorithms, while the computational iterations of the PSEM algorithm are fewer than or equal to other algorithms. Finally, we see that both computational times and iterations of the CSEM and PSEM algorithms are better than or equal to those of the PHMEM algorithm.

Remark 3.

Let us consider the case of parameters ${\alpha }_k=1$ and ${\beta }_k=0$, in which the Ishikawa iteration will be reduced to the Picard iteration. We notice that the convergence of PHMEM algorithm cannot be guaranteed in this setting. The computational results of the CSEM and PSEM algorithms are shown as follows.

From

Table 2. Numerical results for parameters ${\alpha }_k=1$ and ${\beta }_k=0$.

Average Times (sec)		Average Iterations
CSEM	PSEM	CSEM	PSEM
4.657696	137.200812	12.50	11.50

, we see that both computational times and iterations are better than all those cases presented in Table 1. However, it should be warned that the Picard iteration method may not always converge to a fixed point of a nonexpansive mapping in general. For example, see [43].

5. Conclusions

We introduce the methods for finding a common element of the set of fixed points of a finite family for quasi-nonexpansive mappings and the solution set of equilibrium problems of a finite family for pseudomonotone bifunctions in a real Hilbert space. In fact, we consider both extragradient and shrinking projection methods together in combination with Ishikawa’s iteration concept for introducing a sequence which is strongly convergent to a common solution of the considered problems. Some numerical experiments are also provided and discussed. For the future research direction, the convergence analysis of the proposed algorithms and some practical applications should be considered and implemented.