Inertial Extragradient Methods for Solving Split Equilibrium Problems

Abstract

This paper presents two inertial extragradient algorithms for finding a solution of split pseudomonotone equilibrium problems in the setting of real Hilbert spaces. The weak and strong convergence theorems of the introduced algorithms are presented under some constraint qualifications of the scalar sequences. The discussions on the numerical experiments are also provided to demonstrate the effectiveness of the proposed algorithms.

1. Introduction

The equilibrium problem is a 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 $f:H\times H\to \mathbb{R}$ is a bifunction. The solution set of the equilibrium problem (1) will be represented by $EP(f,C)$. It is well known that 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). It is pointed out that one of the most popular methods for solving the equilibrium problem (1), when f is a monotone bifunction, is the proximal point method (see [5]). However, the proximal point method cannot be guaranteed for a weaker assumption, such as f is a pseudomonotone bifunction. To overcome this drawback, Tran et al. [6] proposed the following so-called extragradient method for solving the equilibrium problem, when the bifunction f is pseudomonotone and satisfies Lipschitz-type continuous conditions with positive constants $c_1$ and $c_2$:

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

(2)

where $0<\lambda <min\left\{\frac{1}{2c_1},\frac{1}{2c_2}\right\}$. They proved that the sequence $\left\{x_k\right\}$ generated by (2) converges weakly to a solution of the equilibrium problem (1).

Meanwhile, the inertial-type methods have received a lot of attention from many researchers. This method originates from an implicit discretization method (heavy ball method) of the second-order dynamical in time [7,8] and can be regarded as a method of speeding up the convergence properties. In general, the main feature of the inertial-type methods is that the next iterate is constructed by the two previous iterates. The inertial techniques have been proposed for solving the equilibrium problems, for instance, [9,10] and the references therein. In 2019, by using the ideas of inertial and extragradient methods, Vinh and Muu [11] proposed the following method for solving the equilibrium problem, when the bifunction f is pseudomonotone and satisfies Lipschitz-type continuous conditions with positive constants $c_1$ and $c_2$:

$$\left\{\begin{array}{c}{x}_0,x_1\in C,\hfill \\ w_k=x_k+{\theta }_k(x_k-x_{k-1}),\hfill \\ y_k=argmin\left\{\lambda f(w_k,y)+\frac{1}{2}{\parallel y-w_k\parallel }^2:y\in C\right\},\hfill \\ x_{k+1}=argmin\left\{\lambda f(y_k,y)+\frac{1}{2}{\parallel y-w_k\parallel }^2:y\in C\right\},\hfill \end{array}\right.$$

(3)

where $0<\lambda <min\left\{\frac{1}{2c_1},\frac{1}{2c_2}\right\}$ and ${\theta }_k$ is a suitable parameter. They proved that the sequence $\left\{x_k\right\}$ generated by (3) converges weakly to a solution of the equilibrium problem (1). Observe that, in the case of ${\theta }_k=0$, for all $k\in \mathbb{N}$, the algorithm (3) is nothing but the algorithm (2). Moreover, in [11], the authors proposed the following method, when the bifunction f is pseudomonotone and satisfies Lipschitz-type continuous conditions with positive constants $c_1$ and $c_2$:

$$\left\{\begin{array}{c}{x}_0,x_1\in C,\hfill \\ w_k=x_k+{\theta }_k(x_k-x_{k-1}),\hfill \\ y_k=argmin\left\{\lambda f(w_k,y)+\frac{1}{2}{\parallel y-w_k\parallel }^2:y\in C\right\},\hfill \\ z_k=argmin\left\{\lambda f(y_k,y)+\frac{1}{2}{\parallel y-w_k\parallel }^2:y\in C\right\},\hfill \\ x_{k+1}=(1-{\beta }_k-{\gamma }_k)w_k+{\beta }_k{z}_k,\hfill \end{array}\right.$$

(4)

where $0<\lambda <min\left\{\frac{1}{2c_1},\frac{1}{2c_2}\right\}$, $\left\{{\beta }_k\right\},\left\{{\gamma }_k\right\}\subset (0,1)$ such that ${\displaystyle \sum _{k=0}^{\infty }}{\gamma }_k=\infty$, $\underset{k\to \infty }{lim}{\gamma }_k=0$, $\underset{k\to \infty }{inf}{\beta }_k(1-{\beta }_k-{\gamma }_k)>0$, and ${\theta }_k$ is a suitable parameter. They proved that the sequence $\left\{x_k\right\}$ generated by (4) converges strongly to the minimum-norm element of the solution of the equilibrium problem (1).

On the other hand, Censer and Elfving [12] proposed the following split feasibility problems:

$$\mathrm{Find}{x}^{*}\in C\mathrm{such}\mathrm{that}A{x}^{*}\in Q,$$

(5)

where C and Q are two nonempty closed convex subsets of the real Hilbert spaces $H_1$ and $H_2$, respectively, and $A:H_1\to H_2$ is a bounded linear operator. Many important problems arising from real-world problems can be formulated as the split feasibility problems which had been used for studying signal processing, medical image reconstruction, intensity-modulated radiation therapy, sensor networks, and data compression (see [12,13,14,15] and the references therein).

In 2012, He [16] (see also Moudafi [17]) introduced the split equilibrium problems, as the generalization of the split feasibility problems (5), as follows:

$$\left\{\begin{array}{c}\mathrm{Find}{x}^{*}\in C\mathrm{such}\mathrm{that}f(x^{*},y)\ge 0,\forall y\in C,\hfill \\ \mathrm{and}{u}^{*}:=A{x}^{*}\in Q\mathrm{solves}g(u^{*},v)\ge 0,\forall v\in Q,\hfill \end{array}\right.$$

(6)

where C, Q are two nonempty closed convex subsets of the real Hilbert spaces $H_1$ and $H_2$, respectively, $f:C\times C\to \mathbb{R}$ and $g:Q\times Q\to \mathbb{R}$ are bifunctions, and $A:H_1\to H_2$ is a bounded linear operator. To solve the split equilibrium problems (6), He [16] proposed the following proximal point method, when the bifunctions f and g are monotone:

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ y_k\in C\mathrm{such}\mathrm{that}f(y_k,y)+\frac{1}{r_k}\langle y-y_k,y_k-x_k\rangle \ge 0,\forall y\in C,\hfill \\ u_k\in Q\mathrm{such}\mathrm{that}g(u_k,v)+\frac{1}{r_k}\langle v-u_k,u_k-A{y}_k\rangle \ge 0,\forall v\in Q,\hfill \\ x_{k+1}=P_C(y_k+\eta A^{*}(u_k-A{y}_k)),\hfill \end{array}\right.$$

(7)

where $\eta \in (0,1/\parallel A{\parallel }^2)$, $\left\{r_k\right\}\subset (0,+\infty )$ with $\underset{k\to \infty }{lim\; inf}{r}_k>0$, and $A^{*}$ is the adjoint operator of A. He proved that the sequence $\left\{x_k\right\}$ generated by (7) converges weakly to a solution of the split equilibrium problems (6). Here, the algorithm (7) will be called the PPA Algorithm. After that, under the setting of $f:H_1\times H_1\to \mathbb{R}$ and $g:H_2\times H_2\to \mathbb{R}$, Kim and Dinh [18] proposed the following the extragradient method for finding a solution of the split equilibrium problems, when the bifunctions f and g are pseudomonotone and satisfy Lipschitz-type continuous conditions with positive constants $c_1$ and $c_2$:

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ y_k=argmin\left\{{\lambda }_{k}f(x_k,y)+\frac{1}{2}{\parallel y-x_k\parallel }^2:y\in C\right\},\hfill \\ z_k=argmin\left\{{\lambda }_{k}f(y_k,y)+\frac{1}{2}{\parallel y-x_k\parallel }^2:y\in C\right\},\hfill \\ u_k=argmin\left\{{\mu }_{k}g(A{z}_k,u)+\frac{1}{2}{\parallel u-A{z}_k\parallel }^2:u\in Q\right\},\hfill \\ v_k=argmin\left\{{\mu }_{k}g(u_k,u)+\frac{1}{2}{\parallel u-A{z}_k\parallel }^2:u\in Q\right\},\hfill \\ x_{k+1}=P_C(z_k+\eta A^{*}(v_k-A{z}_k)),\hfill \end{array}\right.$$

(8)

where $\eta \in (0,1/\parallel A{\parallel }^2)$, and $\left\{{\lambda }_k\right\}$, $\left\{{\mu }_k\right\}\subset [\underline{\rho },\overline{\rho }]$ with $0<\underline{\rho }\le \overline{\rho }<min\left\{\frac{1}{2c_1},\frac{1}{2c_2}\right\}$. They proved that the sequence $\left\{x_k\right\}$ generated by (8) converges weakly to a solution of the split equilibrium problems. Here, the algorithm (8) will be called the PEA Algorithm. We point out that the algorithm (8) cannot be applied for solving the problem (6) under the setting of $g:Q\times Q\to \mathbb{R}$, since we can not guarantee if $A{z}_k$ belongs to the considered closed convex set Q.

In this paper, we will continue developing methods for solving the split equilibrium problems (6). That is, some new iterative algorithms will be introduced for finding the solutions of the split equilibrium problems, when the considered bifunctions are pseudomonotone. Some numerical examples and comparison of the introduced methods with the aforesaid algorithms will be discussed.

This paper is organized as follows: In Section 2, some definitions and properties will be reviewed for use in subsequent sections. Section 3 will present two inertial extragradient algorithms and prove their convergence theorems. In Section 4, we will discuss the performance of the two introduced algorithms by comparing to the well-known algorithms.

2. Preliminaries

This section will present the definitions and some important basic properties that will be used in this paper. Let H be a real Hilbert space with inner product $\langle ·,·\rangle$, and its corresponding $\parallel ·\parallel$. The symbols → and ⇀ will be denoted for the strong convergence and the weak convergence in H, respectively.

First, we will recall definitions and facts for concerning the equilibrium problems.

Definition 1

([1,3,19]). Let C be a nonempty closed convex subset of H. A bifunction $f:H\times H\to \mathbb{R}$ is said to be:

(i) 

monotone on C if

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

(ii) 

pseudomonotone on C if

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

(iii) 

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

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

Remark 1.

A monotone bifunction is a pseudomonotone bifunction, but the converse is not true in general, for instance, see [20].

For a nonempty closed convex subset C of H and a bifunction $f:H\times H\to \mathbb{R}$, we are concerned with the following assumptions in this paper:

Assumption 1.

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

Assumption 2.

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

Assumption 3.

f is psuedomonotone on C and $f(x,x)=0$, for each $x\in C$.

Assumption 4.

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

Remark 2.

We note that the solution set $EP(f,C)$ is closed and convex, when the bifunction f satisfies the Assumptions 1–3 (see [6,21,22] for more detail).

The following lemma is important in order to obtain the main results of this paper.

Lemma 1.

([23]) Let $f:H\times H\to \mathbb{R}$ be satisfied Assumptions 2–4. Assume that $EP(f,C)$ is a nonempty set and $0<{\lambda }_0<min\left\{\frac{1}{2L_1},\frac{1}{2L_2}\right\}$. Let $x_0\in H$. If $y_0$ and $z_0$ are constructed by

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

then,

(i) 

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

(ii) 

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

Next, we recall some basic facts in the functional analysis which will be used in the sequel. For a Hilbert space H, we know that

$${\parallel x+y\parallel }^2\le {\parallel x\parallel }^2+2\langle y,x+y\rangle ,$$

(9)

and

$${\parallel \alpha x+\beta y+\gamma z\parallel }^2={\alpha \parallel x\parallel }^2+{\beta \parallel y\parallel }^2+{\gamma \parallel z\parallel }^2{-\alpha \beta \parallel x-y\parallel }^2{-\beta \gamma \parallel y-z\parallel }^2-\alpha \gamma {\parallel x-z\parallel }^2,$$

(10)

for each $x,y,z\in H$, and for each $\alpha ,\beta ,\gamma \in [0,1]$ with $\alpha +\beta +\gamma =1$ (see [11]).

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

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

Lemma 2.

(see [24,25]) 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) 

$P_C$ is a nonexpansive operator, that is,

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

For a function $g:H\to \mathbb{R}$, the subdifferential of g at $z\in H$ is defined by

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

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

Lemma 3.

(see [24]) For any $z\in H$, the subdifferential $\partial g\left(z\right)$ of a continuous convex function g is a nonempty, weakly closed, and bounded convex set.

We end this section by recalling some auxiliary results for obtaining the convergence theorems.

Lemma 4.

([26]) Let H be a Hilbert space and $\left\{x_k\right\}$ a sequence in H such that there exists a nonempty set $S\subset H$ satisfying

(i) 

For each $z\in S$, $\underset{k\to \infty }{lim}\parallel x_k-z\parallel$ exists;

(ii) 

${\omega }_w\left(x_k\right)\subset S$, where ${\omega }_w\left(x_k\right)=\{x\in H:\mathit{there}\mathit{is}a\mathit{subsequence}\left\{x_{k_n}\right\}\mathit{of}\left\{x_k\right\}\mathit{such}\mathit{that}{x}_{k_n}\rightharpoonup x\}$.

Then, there exists $x^{*}\in S$ such that the sequence $\left\{x_k\right\}$ converges weakly to $x^{*}$.

Lemma 5.

([27]) Let $\left\{a_k\right\}$ and $\left\{b_k\right\}$ be sequences of non-negative real numbers such that $a_{k+1}\le a_k+b_k,\forall k\in \mathbb{N}$. If ${\displaystyle \sum _{k=1}^{\infty }}{b}_k<\infty$, then $\underset{k\to \infty }{lim}{a}_k$ exists.

Lemma 6.

([28,29]) Let $\left\{a_k\right\}$ and $\left\{c_k\right\}$ be sequences of non-negative real numbers such that

$$a_{k+1}\le (1-{\delta }_k)a_k+b_k+c_k,\forall k\in \mathbb{N}\cup \left\{0\right\},$$

where $\left\{{\delta }_k\right\}$ is a sequence in $(0,1)$ and $\left\{b_k\right\}$ is a sequence in $\mathbb{R}$. Assume ${\displaystyle \sum _{k=0}^{\infty }}{c}_k<\infty$. Then the following results hold:

(i) 

If there is $M>0$ such that $b_k\le {\delta }_{k}M$, for all $k\in \mathbb{N}\cup \left\{0\right\}$, then $\left\{a_k\right\}$ is a bounded sequence;

(ii) 

If ${\displaystyle \sum _{k=0}^{\infty }}{\delta }_k=\infty$ and $\underset{k\to \infty }{lim\; sup}(b_k/{\delta }_k)\le 0$, then $\underset{k\to \infty }{lim}{a}_k=0$.

Lemma 7.

([30]) Let $\left\{a_k\right\}$ be a sequence of real numbers such that there exists a subsequence $\left\{a_{k_i}\right\}$ of $\left\{a_k\right\}$ such that $a_{k_i}<a_{k_i+1}$, for all $i\in \mathbb{N}$. Then, there exists a nondecreasing sequence $\left\{m_n\right\}$ of positive integers such that $\underset{n\to \infty }{lim}{m}_n=\infty$ and the following properties hold:

$$\begin{array}{c}\hfill a_{m_n}\le a_{m_n+1}\mathit{and}{a}_n\le a_{m_n+1},\end{array}$$

for all (sufficiently large) numbers $n\in \mathbb{N}$. Indeed, $m_n$ is the largest number k in the set $\{1,2,\dots ,n\}$ such that

$$a_k<a_{k+1}.$$

3. Main Results

Let $H_1$ and $H_2$ be two real Hilbert spaces and C and Q be nonempty closed convex subsets of $H_1$ and $H_2$, respectively. Suppose that $f:H_1\times H_1\to \mathbb{R}$ and $g:H_2\times H_2\to \mathbb{R}$ are bifunctions which satisfy Assumptions 1–4 with some positive constants $\{c_1,c_2\}$ and $\{d_1,d_2\}$, respectively. Let us recall the split equilibrium problems:

$$\left\{\begin{array}{c}\mathrm{Find}{x}^{*}\in C\mathrm{such}\mathrm{that}f(x^{*},y)\ge 0,\forall y\in C,\hfill \\ \mathrm{and}{u}^{*}:=A{x}^{*}\in Q\mathrm{solves}g(u^{*},v)\ge 0,\forall v\in Q,\hfill \end{array}\right.$$

(11)

where $f:H_1\times H_1\to \mathbb{R}$ and $g:H_2\times H_2\to \mathbb{R}$ are bifunctions, and $A:H_1\to H_2$ is a bounded linear operator with its adjoint operator $A^{*}$. From now on, the solution set of problem (11) will be denoted by $\mathsf{\Omega }$. That is,

$$\begin{array}{c}\hfill \mathsf{\Omega }:=EP(f,C)\cap A^{-1}\left(EP(g,Q)\right).\end{array}$$

3.1. Inertial Extragradient Method

Now, we introduce Algorithm 1 for solving the split equilibrium problems (11).

Algorithm 1: Inertial Extragradient Method (IEM)

Initialization. Choose parameters $\alpha \in [0,1)$, $\eta \in \left(0,\frac{1}{{\parallel A\parallel }^2}\right)$, $\left\{{\lambda }_k\right\}$ with $0<inf{\lambda }_k\le sup{\lambda }_k<min\left\{\frac{1}{2c_1},\frac{1}{2c_2}\right\}$, $\left\{{\mu }_k\right\}$ with $0<inf{\mu }_k\le sup{\mu }_k<min\left\{\frac{1}{2d_1},\frac{1}{2d_2}\right\}$, and $\left\{{ϵ}_k\right\}\subset [0,\infty )$ such that ${\displaystyle \sum _{k=0}^{\infty }}{ϵ}_k<\infty$. Pick $x_0,x_1\in C$ and set $k=1$. Step 1. Choose ${\theta }_k$ such that $0\le {\theta }_k\le {\overline{\theta }}_k$, where $${\overline{\theta }}_k=\left\{\begin{array}{c}min\left\{\alpha ,\frac{{ϵ}_k}{\parallel x_k-x_{k-1}\parallel }\right\},\mathrm{if}{x}_k\ne x_{k-1},\hfill \\ \alpha ,\mathrm{otherwise},\hfill \end{array}\right.$$ and compute $$w_k=x_k+{\theta }_k(x_k-x_{k-1}).$$ Step 2. Solve the strongly convex program $$\begin{array}{ccc}\hfill & & y_k=argmin\left\{{\lambda }_{k}f(w_k,y)+\frac{1}{2}{\parallel y-w_k\parallel }^2:y\in C\right\}.\hfill \end{array}$$ Step 3. Solve the strongly convex program $$\begin{array}{ccc}\hfill & & z_k=argmin\left\{{\lambda }_{k}f(y_k,y)+\frac{1}{2}{\parallel y-w_k\parallel }^2:y\in C\right\}.\hfill \end{array}$$ Step 4. Solve the strongly convex program $$\begin{array}{ccc}\hfill & & u_k=argmin\left\{{\mu }_{k}g(A{z}_k,u)+\frac{1}{2}{\parallel u-A{z}_k\parallel }^2:u\in Q\right\}.\hfill \end{array}$$ Step 5. Solve the strongly convex program $$\begin{array}{ccc}\hfill & & v_k=argmin\left\{{\mu }_{k}g(u_k,u)+\frac{1}{2}{\parallel u-A{z}_k\parallel }^2:u\in Q\right\}.\hfill \end{array}$$ Step 6. The next approximation $x_{k+1}$ is defined by $$x_{k+1}=P_C(z_k+\eta A^{*}(v_k-A{z}_k)).$$ Step 7. Put $k:=k+1$ and go to Step 1.

Remark 3.

We pointed out that the term ${\theta }_k(x_k-x_{k-1})$, which is included in the IEM Algorithm, is intended to speed up convergence properties and is called the inertial effect. We emphasize that the choice of parameter ${\theta }_k$ may lead to the superior numerical behavior of the IEM Algorithm. Moreover, we observe that if ${\theta }_k=0$, for each $k\in \mathbb{N}$, then the IEM Algorithm reduces to the PEA Algorithm (8), which was presented in [18].

Theorem 1.

Suppose that the solution set Ω is nonempty. Then, the sequence $\left\{x_k\right\}$ which is generated by the IEM Algorithm converges weakly to an element of Ω.

Proof. 

Let $p\in \mathsf{\Omega }$. That is, $p\in EP(f,C)$, and $Ap\in EP(g,Q)$. Then, by Lemma 1 (ii), we have

$$\parallel z_k{-p\parallel }^2\le \parallel w_k{-p\parallel }^2-(1-2{\lambda }_k{c}_1)\parallel w_k-y_k{\parallel }^2-(1-2{\lambda }_k{c}_2){\parallel y_k-z_k\parallel }^2.$$

(12)

This implies that

$$\parallel z_k-p\parallel \le \parallel w_k-p\parallel .$$

(13)

By the definition of $w_k$, we have

$$\begin{array}{ccc}\hfill \parallel w_k{-p\parallel }^2& =& \parallel (1+{\theta }_k)(x_k-p)-{\theta }_k(x_{k-1}-p){\parallel }^2\hfill \\ \hfill & =& (1+{\theta }_k)\parallel x_k{-p\parallel }^2-{\theta }_k\parallel x_{k-1}{-p\parallel }^2+{\theta }_k(1+{\theta }_k){\parallel x_k-x_{k-1}\parallel }^2\hfill \\ & \le & (1+{\theta }_k)\parallel x_k{-p\parallel }^2-{\theta }_k\parallel x_{k-1}{-p\parallel }^2+2{\theta }_k{\parallel x_k-x_{k-1}\parallel }^2.\hfill \end{array}$$

(14)

Thus, in view of (12) and(14), we obtain

$$\begin{array}{ccc}\hfill \parallel z_k{-p\parallel }^2-{\parallel x_k-p\parallel }^2& \le & {\theta }_k(\parallel x_k{-p\parallel }^2-\parallel x_{k-1}{-p\parallel }^2)+2{\theta }_k{\parallel x_k-x_{k-1}\parallel }^2\hfill \\ & & -(1-2{\lambda }_k{c}_1)\parallel w_k-y_k{\parallel }^2-(1-2{\lambda }_k{c}_2){\parallel y_k-z_k\parallel }^2.\hfill \end{array}$$

(15)

On the other hand, by Lemma 1 (ii), we obtain

$$\parallel v_k{-Ap\parallel }^2\le \parallel A{z}_k{-Ap\parallel }^2-(1-2{\mu }_k{d}_1)\parallel A{z}_k-u_k{\parallel }^2-(1-2{\mu }_k{d}_2){\parallel u_k-v_k\parallel }^2.$$

(16)

This implies that

$$\parallel v_k-Ap\parallel \le \parallel A{z}_k-Ap\parallel .$$

(17)

By the definition of $x_{k+1}$ and the nonexpansivity of $P_C$, we have

$$\begin{array}{ccc}\hfill \parallel x_{k+1}{-p\parallel }^2& \le & \parallel (z_k-p)+\eta A^{*}(v_k-A{z}_k){\parallel }^2\hfill \\ & =& \parallel z_k{-p\parallel }^2+{\eta }^2{\parallel A\parallel }^2{\parallel v_k-A{z}_k\parallel }^2+2\eta \langle A(z_k-p),v_k-A{z}_k\rangle .\hfill \end{array}$$

(18)

Consider,

$$\begin{array}{ccc}\hfill 2\langle A(z_k-p),v_k-A{z}_k\rangle & =& 2\langle v_k-Ap,v_k-A{z}_k\rangle -2{\parallel v_k-A{z}_k\parallel }^2\hfill \\ & =& \parallel v_k{-Ap\parallel }^2-\parallel v_k-A{z}_k{\parallel }^2-{\parallel A{z}_k-Ap\parallel }^2.\hfill \end{array}$$

(19)

Using this one together with (18), we obtain

$$\begin{array}{ccc}\hfill \parallel x_{k+1}{-p\parallel }^2& \le & \parallel z_k{-p\parallel }^2{-\eta (1-\eta \parallel A\parallel }^2)\parallel v_k-A{z}_k{\parallel }^2\hfill \\ & & +\eta (\parallel v_k-Ap{\parallel }^2-\parallel A{z}_k-Ap{\parallel }^2).\hfill \end{array}$$

(20)

Combining with (17) implies that

$$\begin{array}{ccc}\hfill \parallel x_{k+1}{-p\parallel }^2& \le & \parallel z_k{-p\parallel }^2{-\eta (1-\eta \parallel A\parallel }^2)\parallel v_k-A{z}_k{\parallel }^2.\hfill \end{array}$$

(21)

Thus, by the choice of $\eta$, we have

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-p\parallel & \le & \parallel z_k-p\parallel .\hfill \end{array}$$

(22)

Now, the relations (13) and (22) imply that

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-p\parallel & \le & \parallel w_k-p\parallel .\hfill \end{array}$$

(23)

So, it follows from the definition of $w_k$ that

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-p\parallel & \le & \parallel x_k-p\parallel +{\theta }_k\parallel x_k-x_{k-1}\parallel .\hfill \end{array}$$

(24)

Due to the properties of the sequences $\left\{{\theta }_k\right\}$ and $\left\{{ϵ}_k\right\}$, we observe that

$${\displaystyle \sum _{k=1}^{\infty }}{\theta }_k\parallel x_k-x_{k-1}\parallel \le {\displaystyle \sum _{k=1}^{\infty }}{\overline{\theta }}_k\parallel x_k-x_{k-1}\parallel \le {\displaystyle \sum _{k=1}^{\infty }}{ϵ}_k<\infty .$$

(25)

Then, by (24), (25), and Lemma 5, we show $\underset{k\to \infty }{lim}\parallel x_k-p\parallel$ exists. Consequently, the sequence $\left\{x_k\right\}$ is bounded. In addition, in view of (15) and (22), we see that

$$\begin{array}{ccc}\hfill (1-2{\lambda }_k{c}_1)\parallel w_k-y_k{\parallel }^2+(1-2{\lambda }_k{c}_2){\parallel y_k-z_k\parallel }^2& \le & \parallel x_k{-p\parallel }^2-\parallel x_{k+1}{-p\parallel }^2+{\theta }_k(\parallel x_k{-p\parallel }^2\hfill \\ & & -\parallel x_{k-1}{-p\parallel }^2)+2{\theta }_k{\parallel x_k-x_{k-1}\parallel }^2.\hfill \end{array}$$

(26)

Thus, by the choices of the control sequences $\left\{{\lambda }_k\right\}$ together with the existence of $\underset{k\to \infty }{lim}\parallel x_k-p\parallel$ and $\underset{k\to \infty }{lim}{\theta }_k\parallel x_k-x_{k-1}\parallel =0$, we have

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

(27)

and

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

(28)

These imply that

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

(29)

Moreover, by using $\underset{k\to \infty }{lim}{\theta }_k\parallel x_k-x_{k-1}\parallel =0$, we have

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

(30)

Using this one together with (27), we obtain

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

(31)

Thus, it follows from (28) that

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

(32)

On the other hand, by using (21), we see that

$$\begin{array}{ccc}\hfill {\eta (1-\eta \parallel A\parallel }^2)\parallel v_k-A{z}_k{\parallel }^2& \le & \parallel z_k{-p\parallel }^2-{\parallel x_{k+1}-p\parallel }^2\hfill \\ \hfill & \le & (\parallel z_k-x_k\parallel +\parallel x_k-p\parallel -\parallel x_{k+1}-p\parallel \left)\right(\parallel z_k-p\parallel \hfill \\ & & +\parallel x_{k+1}-p\parallel ).\hfill \end{array}$$

(33)

Thus, by the existence of $\underset{k\to \infty }{lim}\parallel x_k-p\parallel$ and (32), we have

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

(34)

Furthermore, in view of (16), we obtain

$$\begin{array}{ccc}\hfill (1-2{\mu }_k{d}_1)\parallel A{z}_k-u_k{\parallel }^2+(1-2{\mu }_k{d}_2){\parallel u_k-v_k\parallel }^2& \le & \parallel A{z}_k{-Ap\parallel }^2-{\parallel v_k-Ap\parallel }^2\hfill \\ \hfill & =& \parallel A{z}_k-v_k\parallel (\parallel A{z}_k-Ap\parallel \hfill \\ \hfill & & +\parallel v_k-Ap\parallel ).\hfill \end{array}$$

Thus, applying (34) to the above inequality, we have

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

(35)

and

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

(36)

Now, we will complete the proof of this theorem by using Lemma 4. Notice that, it remains to show ${\omega }_w\left(x_k\right)\subset \mathsf{\Omega }$. Let $x^{*}\in {\omega }_w\left(x_k\right)$ and $\left\{x_{k_n}\right\}$ be a subsequence of $\left\{x_k\right\}$ such that $x_{k_n}\rightharpoonup x^{*}$, as $n\to \infty$.

We know that, by using (30)–(32), we also have $w_{k_n}\rightharpoonup x^{*}$, $y_{k_n}\rightharpoonup x^{*}$, and $z_{k_n}\rightharpoonup x^{*}$, as $n\to \infty$. The latter fact also implies that $A{z}_{k_n}\rightharpoonup A{x}^{*}$, as $n\to \infty$. Using this one together with (35), we obtain $u_{k_n}\rightharpoonup A{x}^{*}$, as $n\to \infty$. Since C and Q are closed and convex sets, so C and Q are weakly closed, therefore, $x^{*}\in C$ and $A{x}^{*}\in Q$. By Lemma 1 (i), we have

$${\lambda }_{k_n}[f(w_{k_n},y)-f(w_{k_n},y_{k_n})]\ge \langle y_{k_n}-w_{k_n},y_{k_n}-y\rangle ,\forall y\in C,$$

and

$${\mu }_{k_n}[g(A{z}_{k_n},u)-g(A{z}_{k_n},u_{k_n})]\ge \langle u_{k_n}-A{z}_{k_n},u_{k_n}-u\rangle ,\forall u\in Q.$$

These imply that

$$f(w_{k_n},y)-f(w_{k_n},y_{k_n})\ge -{\displaystyle \frac{1}{{\lambda }_{k_n}}}\parallel y_{k_n}-w_{k_n}\parallel \parallel y_{k_n}-y\parallel ,\forall y\in C,$$

and

$$g(A{z}_{k_n},u)-g(A{z}_{k_n},u_{k_n})\ge -{\displaystyle \frac{1}{{\mu }_{k_n}}}\parallel u_{k_n}-A{z}_{k_n}\parallel \parallel u_{k_n}-u\parallel ,\forall u\in Q.$$

Thus, by using (27), (35), and the weak continuity of f and g, we have

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

and

$$g(A{x}^{*},u)\ge 0,\forall u\in Q.$$

Then, we show that $x^{*}\in \mathsf{\Omega }$. This shows that ${\omega }_w\left(x_k\right)\subset \mathsf{\Omega }$. Hence, by Lemma 4, we can conclude that the sequence $\left\{x_k\right\}$ converges weakly to an element of $\mathsf{\Omega }$. This completes the proof. □

3.2. Mann-Type Inertial Extragradient Method

In order to obtain a strong convergence result, we propose Algorithm 2 by using the Mann-type techniques (see [11,31]).

Algorithm 2: Mann-type Inertial Extragradient Method (MIEM)

Initialization. Choose parameters $\alpha \in [0,1)$, $\eta \in \left(0,\frac{1}{{\parallel A\parallel }^2}\right)$, $\left\{{\lambda }_k\right\}$ with $0<inf{\lambda }_k\le sup{\lambda }_k<min\left\{\frac{1}{2c_1},\frac{1}{2c_2}\right\}$, $\left\{{\mu }_k\right\}$ with $0<inf{\mu }_k\le sup{\mu }_k<min\left\{\frac{1}{2d_1},\frac{1}{2d_2}\right\}$, and $\left\{{ϵ}_k\right\}\subset [0,\infty )$, $\left\{{\beta }_k\right\}\subset (0,1)$, $\left\{{\gamma }_k\right\}\subset (0,1)$ such that $\underset{k\to \infty }{inf}{\beta }_k(1-{\beta }_k-{\gamma }_k)>0$, ${\displaystyle \sum _{k=0}^{\infty }}{\gamma }_k=\infty$, $\underset{k\to \infty }{lim}{\gamma }_k=0$, ${\displaystyle \sum _{k=0}^{\infty }}{ϵ}_k<\infty$, and ${ϵ}_k=o\left({\gamma }_k\right)$, where ${ϵ}_k=o\left({\gamma }_k\right)$ means that the sequence $\left\{{ϵ}_k\right\}$ is an infinitesimal of higher order than $\left\{{\gamma }_n\right\}$. Pick $x_0,x_1\in C$ and set $k=1$. Step 1. Choose ${\theta }_k$ such that $0\le {\theta }_k\le {\overline{\theta }}_k$, where $${\overline{\theta }}_k=\left\{\begin{array}{c}min\left\{\alpha ,\frac{{ϵ}_k}{\parallel x_k-x_{k-1}\parallel }\right\},\mathrm{if}{x}_k\ne x_{k-1},\hfill \\ \alpha ,\mathrm{otherwise},\hfill \end{array}\right.$$ and compute $$w_k=x_k+{\theta }_k(x_k-x_{k-1}).$$ Step 2. Solve the strongly convex program $$\begin{array}{ccc}\hfill & & y_k=argmin\left\{{\lambda }_{k}f(w_k,y)+\frac{1}{2}{\parallel y-w_k\parallel }^2:y\in C\right\}.\hfill \end{array}$$ Step 3. Solve the strongly convex program $$\begin{array}{ccc}\hfill & & z_k=argmin\left\{{\lambda }_{k}f(y_k,y)+\frac{1}{2}{\parallel y-w_k\parallel }^2:y\in C\right\}.\hfill \end{array}$$ Step 4. Solve the strongly convex program $$\begin{array}{ccc}\hfill & & u_k=argmin\left\{{\mu }_{k}g(A{z}_k,u)+\frac{1}{2}{\parallel u-A{z}_k\parallel }^2:u\in Q\right\}.\hfill \end{array}$$ Step 5. Solve the strongly convex program $$\begin{array}{ccc}\hfill & & v_k=argmin\left\{{\mu }_{k}g(u_k,u)+\frac{1}{2}{\parallel u-A{z}_k\parallel }^2:u\in Q\right\}.\hfill \end{array}$$ Step 6. Compute $$t_k=P_C(z_k+\eta A^{*}(v_k-A{z}_k)).$$ . Step 7. The next approximation $x_{k+1}$ is defined by $$x_{k+1}=(1-{\beta }_k-{\gamma }_k)w_k+{\beta }_k{t}_k.$$ . Step 8. Put $k:=k+1$ and go to Step 1.

Theorem 2.

Suppose that the solution set Ω is nonempty. Then, the sequence $\left\{x_k\right\}$ which is generated by the MIEM Algorithm converges strongly to the minimum-norm element of Ω.

Proof. 

Let $p\in \mathsf{\Omega }$. That is, $p\in EP(f,C)$, and $Ap\in EP(g,Q)$. Following the proof of Theorem 1, we can check that

$$\parallel z_k-p\parallel \le \parallel w_k-p\parallel ,$$

(37)

$$\parallel t_k-p\parallel \le \parallel z_k-p\parallel ,$$

(38)

$$\parallel v_k-Ap\parallel \le \parallel A{z}_k-Ap\parallel ,$$

(39)

$$\begin{array}{ccc}\hfill \parallel w_k{-p\parallel }^2& \le & \parallel x_k{-p\parallel }^2+{\theta }_k(\parallel x_k{-p\parallel }^2-\parallel x_{k-1}{-p\parallel }^2)+2{\theta }_k{\parallel x_k-x_{k-1}\parallel }^2,\hfill \end{array}$$

(40)

and

$$\begin{array}{ccc}\hfill \parallel t_k{-p\parallel }^2& \le & \parallel z_k{-p\parallel }^2{-\eta (1-\eta \parallel A\parallel }^2)\parallel v_k-A{z}_k{\parallel }^2.\hfill \end{array}$$

(41)

By the definition of $x_{k+1}$, we obtain

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-p\parallel & =& \parallel (1-{\beta }_k-{\gamma }_k)(w_k-p)+{\beta }_k(t_k-p)-{\gamma }_{k}p\parallel \hfill \\ \hfill & \le & (1-{\beta }_k-{\gamma }_k)\parallel w_k-p\parallel +{\beta }_k\parallel t_k-p\parallel +{\gamma }_k\parallel p\parallel .\hfill \end{array}$$

It follows from (37) and (38) that

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-p\parallel & \le & (1-{\gamma }_k)\parallel w_k-p\parallel +{\gamma }_k\parallel p\parallel .\hfill \end{array}$$

Thus, by the definition of $w_k$, we have

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-p\parallel & \le & (1-{\gamma }_k)\parallel x_k-p\parallel +(1-{\gamma }_k){\theta }_k\parallel x_k-x_{k-1}\parallel +{\gamma }_k\parallel p\parallel \hfill \\ & =& (1-{\gamma }_k)\parallel x_k-p\parallel +{\gamma }_k({\sigma }_k+\parallel p\parallel ),\hfill \end{array}$$

(42)

where ${\sigma }_k=(1-{\gamma }_k){\displaystyle \frac{{\theta }_k}{{\gamma }_k}}\parallel x_k-x_{k-1}\parallel$. Due to the choices of the sequence $\left\{{\theta }_k\right\}$, we obtain that

$${\sigma }_k=(1-{\gamma }_k){\displaystyle \frac{{\theta }_k}{{\gamma }_k}}\parallel x_k-x_{k-1}\parallel \le (1-{\gamma }_k){\displaystyle \frac{{ϵ}_k}{{\gamma }_k}}.$$

Thus, by the properties of ${ϵ}_k=o\left({\gamma }_k\right)$ and $\underset{k\to \infty }{lim}{\gamma }_k=0$, we have

$$\underset{k\to \infty }{lim}{\sigma }_k=0.$$

(43)

This implies that the sequence $\left\{{\sigma }_k\right\}$ is a null sequence. Put $M=max\{\parallel p\parallel ,\underset{k\in \mathbb{N}}{sup}{\sigma }_k\}$. Then, by (42) and Lemma 6 (i), the sequence $\{\parallel x_k-p\parallel \}$ is bounded. Consequently, $\left\{x_k\right\}$ is a bounded sequence.

In addition, by the definition of $x_{k+1}$ and (10), we have

$$\begin{array}{ccc}\hfill \parallel x_{k+1}{-p\parallel }^2& =& \parallel (1-{\beta }_k-{\gamma }_k)(w_k-p)+{\beta }_k(t_k-p)+{\gamma }_k(-p){\parallel }^2\hfill \\ \hfill & \le & (1-{\beta }_k-{\gamma }_k)\parallel w_k{-p\parallel }^2+{\beta }_k\parallel t_k{-p\parallel }^2+{\gamma }_k{\parallel p\parallel }^2\hfill \\ & & -{\beta }_k(1-{\beta }_k-{\gamma }_k){\parallel w_k-t_k\parallel }^2.\hfill \end{array}$$

(44)

Thus, by using (38), (40), and Lemma 1 (ii), we obtain that

$$\begin{array}{ccc}\hfill \parallel x_{k+1}{-p\parallel }^2& \le & (1-{\beta }_k-{\gamma }_k)\parallel w_k{-p\parallel }^2+{\beta }_k\parallel z_k{-p\parallel }^2+{\gamma }_k{\parallel p\parallel }^2\hfill \\ \hfill & & -{\beta }_k(1-{\beta }_k-{\gamma }_k){\parallel w_k-t_k\parallel }^2\hfill \\ \hfill & \le & (1-{\gamma }_k)\parallel w_k{-p\parallel }^2-{\beta }_k(1-{\beta }_k-{\gamma }_k)\parallel w_k-t_k{\parallel }^2+{\gamma }_k{\parallel p\parallel }^2\hfill \\ \hfill & & -{\beta }_k(1-2{\lambda }_k{c}_1)\parallel w_k-y_k{\parallel }^2-{\beta }_k(1-2{\lambda }_k{c}_2){\parallel y_k-z_k\parallel }^2\hfill \\ \hfill & \le & (1-{\gamma }_k)\parallel x_k{-p\parallel }^2+{\theta }_k(1-{\gamma }_k)(\parallel x_k{-p\parallel }^2-\parallel x_{k-1}-p{\parallel }^2)\hfill \\ \hfill & & +2{\theta }_k(1-{\gamma }_k)\parallel x_k-x_{k-1}{\parallel }^2-{\beta }_k(1-{\beta }_k-{\gamma }_k)\parallel w_k-t_k{\parallel }^2+{\gamma }_k{\parallel p\parallel }^2\hfill \\ \hfill & & -{\beta }_k(1-2{\lambda }_k{c}_1)\parallel w_k-y_k{\parallel }^2-{\beta }_k(1-2{\lambda }_k{c}_2){\parallel y_k-z_k\parallel }^2.\hfill \end{array}$$

This implies that

$$\begin{array}{ccc}\hfill & & {\beta }_k(1-{\beta }_k-{\gamma }_k)\parallel w_k-t_k{\parallel }^2+{\beta }_k(1-2{\lambda }_k{c}_1)\parallel w_k-y_k{\parallel }^2+{\beta }_k(1-2{\lambda }_k{c}_2){\parallel y_k-z_k\parallel }^2\hfill \\ \hfill & \le & \parallel x_k{-p\parallel }^2-\parallel x_{k+1}{-p\parallel }^2+{\theta }_k(1-{\gamma }_k)(\parallel x_k{-p\parallel }^2-\parallel x_{k-1}-p{\parallel }^2)\hfill \\ & & +2{\theta }_k(1-{\gamma }_k)\parallel x_k-x_{k-1}{\parallel }^2+{\gamma }_k{\parallel p\parallel }^2.\hfill \end{array}$$

(45)

Next, we will show that $\left\{x_k\right\}$ converges strongly to $\tilde{p}:=P_{\mathsf{\Omega }}\left(0\right)$. We consider the following two possible cases.

Case 1. Suppose that there exists $k_0\in \mathbb{N}$ such that $\parallel x_{k+1}-\tilde{p}\parallel \le \parallel x_k-\tilde{p}\parallel$, for all $k\ge k_0$. This means that $\{\parallel x_k-\tilde{p}{\parallel \}}_{k\ge k_0}$ is a nonincreasing sequence. Consequently, by using this one together with the boundness property of $\{\parallel x_k-\tilde{p}\parallel \}$, we have that the limit of $\parallel x_k-\tilde{p}\parallel$ exists. Since $\underset{k\to \infty }{lim}{\theta }_k\parallel x_k-x_{k-1}\parallel =0$ and the properties of the control sequences $\left\{{\beta }_k\right\}$, $\left\{{\gamma }_k\right\}$, $\left\{{\lambda }_k\right\}$, $\left\{{\theta }_k\right\}$, it follows from (45) that

$$\underset{k\to \infty }{lim}\parallel w_k-t_k\parallel =0,$$

(46)

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

(47)

and

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

(48)

These imply that

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

(49)

Moreover, since $\underset{k\to \infty }{lim}{\theta }_k\parallel x_k-x_{k-1}\parallel =0$, we obtain

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

(50)

Using this one together with (47), we obtain

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

(51)

It follows from (48) that

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

(52)

On the other hand, in view of (41), we see that

$$\begin{array}{ccc}\hfill {\eta (1-\eta \parallel A\parallel }^2)\parallel v_k-A{z}_k{\parallel }^2& \le & \parallel z_k-\tilde{p}{\parallel }^2-{\parallel t_k-\tilde{p}\parallel }^2\hfill \\ & \le & \parallel z_k-t_k\parallel (\parallel z_k-\tilde{p}\parallel +\parallel t_k-\tilde{p}\parallel ).\hfill \end{array}$$

Thus, by using (49), we have

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

(53)

Furthermore, by Lemma 1 (ii), we obtain that

$$\begin{array}{ccc}\hfill (1-2{\mu }_k{d}_1)\parallel A{z}_k-u_k{\parallel }^2+(1-2{\mu }_k{d}_2){\parallel u_k-v_k\parallel }^2& \le & \parallel A{z}_k-A\tilde{p}{\parallel }^2-{\parallel v_k-A\tilde{p}\parallel }^2\hfill \\ \hfill & =& \parallel A{z}_k-v_k\parallel (\parallel A{z}_k-A\tilde{p}\parallel \hfill \\ \hfill & & +\parallel v_k-A\tilde{p}\parallel ).\hfill \end{array}$$

Then, applying (53) to the above inequality, we have

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

(54)

and

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

(55)

Now, let $x^{*}\in {\omega }_w\left(x_k\right)$ and $\left\{x_{k_n}\right\}$ be a subsequence of $\left\{x_k\right\}$ such that $x_{k_n}\rightharpoonup x^{*}$, as $n\to \infty$. Following the line proof of Theorem 1, we can show that $x^{*}\in \mathsf{\Omega }$. This means that ${\omega }_w\left(x_k\right)\subset \mathsf{\Omega }$. Put $s_k=(1-{\beta }_k)w_k+{\beta }_k{t}_k$. The relations (37) and (38) imply that

$$\begin{array}{ccc}\hfill \parallel s_k-\tilde{p}\parallel & \le & (1-{\beta }_k)\parallel w_k-\tilde{p}\parallel +{\beta }_k\parallel t_k-\tilde{p}\parallel \hfill \\ & \le & \parallel w_k-\tilde{p}\parallel .\hfill \end{array}$$

(56)

By the definition of $x_{k+1}$, we see that

$$\begin{array}{ccc}\hfill x_{k+1}& =& s_k-{\gamma }_k{w}_k\hfill \\ \hfill & =& (1-{\gamma }_k)s_k-{\gamma }_k(w_k-s_k)\hfill \\ \hfill & =& (1-{\gamma }_k)s_k-{\gamma }_k{\beta }_k(w_k-t_k).\hfill \end{array}$$

It follows from (9) that

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-\tilde{p}{\parallel }^2& =& \parallel (1-{\gamma }_k)(s_k-\tilde{p})-{\gamma }_k{\beta }_k(w_k-t_k)-{\gamma }_k\tilde{p}{\parallel }^2\hfill \\ \hfill & \le & (1-{\gamma }_k){\parallel s_k-\tilde{p}\parallel }^2-2{\gamma }_k{\beta }_k\langle w_k-t_k,x_{k+1}-\tilde{p}\rangle \hfill \\ & & -2{\gamma }_k\langle \tilde{p},x_{k+1}-\tilde{p}\rangle .\hfill \end{array}$$

(57)

Thus, by using (56), we have

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-\tilde{p}{\parallel }^2& \le & (1-{\gamma }_k)\parallel w_k-\tilde{p}{\parallel }^2+{\gamma }_k[-2{\beta }_k\langle w_k-t_k,x_{k+1}-\tilde{p}\rangle \hfill \\ & & +2\langle x_{k+1}-\tilde{p},-\tilde{p}\rangle ].\hfill \end{array}$$

(58)

Consider,

$$\begin{array}{ccc}\hfill \parallel w_k-\tilde{p}{\parallel }^2& \le & (\parallel x_k-\tilde{p}\parallel +{\theta }_k\parallel x_k-x_{k-1}{\parallel )}^2\hfill \\ \hfill & \le & \parallel x_k-\tilde{p}{\parallel }^2+2{\theta }_k\parallel x_k-\tilde{p}\parallel \parallel x_k-x_{k-1}\parallel +{\theta }_k{\parallel x_k-x_{k-1}\parallel }^2\hfill \\ & \le & \parallel x_k-\tilde{p}{\parallel }^2+3M{\theta }_k\parallel x_k-x_{k-1}\parallel ,\hfill \end{array}$$

(59)

where $M=\underset{k\in \mathbb{N}}{sup}\{\parallel x_k-\tilde{p}\parallel ,\parallel x_k-x_{k-1}\parallel \}$. This, together with (58), implies that

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-\tilde{p}{\parallel }^2& \le & (1-{\gamma }_k)\parallel x_k-\tilde{p}{\parallel }^2+3M{\theta }_k(1-{\gamma }_k)\parallel x_k-x_{k-1}\parallel \hfill \\ \hfill & & +{\gamma }_k[-2{\beta }_k\langle w_k-t_k,x_{k+1}-\tilde{p}\rangle +2\langle x_{k+1}-\tilde{p},-\tilde{p}\rangle ]\hfill \\ \hfill & =& (1-{\gamma }_k)\parallel x_k-\tilde{p}{\parallel }^2+{\gamma }_k[3M(1-{\gamma }_k){\displaystyle \frac{{\theta }_k}{{\gamma }_k}}\parallel x_k-x_{k-1}\parallel \hfill \\ & & -2{\beta }_k\langle w_k-t_k,x_{k+1}-\tilde{p}\rangle +2\langle x_{k+1}-\tilde{p},-\tilde{p}\rangle ].\hfill \end{array}$$

(60)

Thus, by the properties of $\tilde{p}:=P_{\mathsf{\Omega }}\left(0\right)$ and $x^{*}\in {\omega }_w\left(x_k\right)\subset \mathsf{\Omega }$, we obtain

$$\underset{k\to \infty }{lim\; sup}\langle x_{k+1}-\tilde{p},-\tilde{p}\rangle =\underset{n\to \infty }{lim}\langle x_{k_n+1}-\tilde{p},-\tilde{p}\rangle =\langle x^{*}-\tilde{p},-\tilde{p}\rangle \le 0.$$

(61)

Hence, by (43), (46), (60), (61), and Lemma 6 (ii), we have

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

(62)

This completes the proof for the first case.

Case 2. Suppose that there exists a subsequence $\{\parallel x_{k_i}-\tilde{p}\parallel \}$ of $\{\parallel x_k-\tilde{p}\parallel \}$ such that

$$\begin{array}{c}\hfill \parallel x_{k_i}-\tilde{p}\parallel <\parallel x_{k_i+1}-\tilde{p}\parallel ,\forall i\in \mathbb{N}.\end{array}$$

According to Lemma 7, there exists a nondecreasing sequence $\left\{m_n\right\}\subset \mathbb{N}$ such that $\underset{n\to \infty }{lim}{m}_n=\infty$, and

$$\begin{array}{c}\hfill \parallel x_{m_n}-\tilde{p}\parallel \le \parallel x_{m_n+1}-\tilde{p}\parallel \mathrm{and}\parallel x_n-\tilde{p}\parallel \le \parallel x_{m_n+1}-\tilde{p}\parallel ,\forall n\in \mathbb{N}.\end{array}$$

(63)

It follows from (45) that

$$\begin{array}{ccc}\hfill & & {\beta }_{m_n}(1-{\beta }_{m_n}-{\gamma }_{m_n})\parallel w_{m_n}-t_{m_n}{\parallel }^2+{\beta }_{m_n}(1-2{\lambda }_{m_n}{c}_1){\parallel w_{m_n}-y_{m_n}\parallel }^2\hfill \\ \hfill & & +{\beta }_{m_n}(1-2{\lambda }_{m_n}{c}_2){\parallel y_{m_n}-z_{m_n}\parallel }^2\hfill \\ \hfill & \le & \parallel x_{m_n}-\tilde{p}{\parallel }^2-\parallel x_{m_n+1}-\tilde{p}{\parallel }^2+{\theta }_{m_n}(1-{\gamma }_{m_n})(\parallel x_{m_n}-\tilde{p}{\parallel }^2-\parallel x_{m_n-1}-\tilde{p}{\parallel }^2)\hfill \\ \hfill & & +2{\theta }_{m_n}(1-{\gamma }_{m_n})\parallel x_{m_n}-x_{m_n-1}{\parallel }^2+{\gamma }_{m_n}{\parallel \tilde{p}\parallel }^2\hfill \\ \hfill & \le & {\theta }_{m_n}(1-{\gamma }_{m_n})\parallel x_{m_n}-x_{m_n-1}\parallel (\parallel x_{m_n}-\tilde{p}\parallel +\parallel x_{m_n-1}-\tilde{p}\parallel )\hfill \\ & & +2{\theta }_{m_n}(1-{\gamma }_{m_n})\parallel x_{m_n}-x_{m_n-1}{\parallel }^2+{\gamma }_{m_n}{\parallel \tilde{p}\parallel }^2.\hfill \end{array}$$

(64)

Following the line proof of Case 1, we can show that

$$\underset{n\to \infty }{lim}\parallel w_{m_n}-t_{m_n}\parallel =0,\underset{n\to \infty }{lim}\parallel x_{m_n}-y_{m_n}\parallel =0,\underset{n\to \infty }{lim}\parallel x_{m_n}-z_{m_n}\parallel =0,$$

(65)

$$\underset{n\to \infty }{lim}\parallel v_{m_n}-A{z}_{m_n}\parallel =0,\underset{n\to \infty }{lim}\parallel A{z}_{m_n}-u_{m_n}\parallel =0,\underset{n\to \infty }{lim}\parallel u_{m_n}-v_{m_n}\parallel =0,$$

(66)

$$\underset{n\to \infty }{lim\; sup}\langle x_{m_n+1}-\tilde{p},-\tilde{p}\rangle \le 0,$$

(67)

and

$$\begin{array}{ccc}\hfill \parallel x_{m_n+1}-\tilde{p}{\parallel }^2& \le & (1-{\gamma }_{m_n})\parallel x_{m_n}-\tilde{p}{\parallel }^2+{\gamma }_{m_n}[3M(1-{\gamma }_{m_n}){\displaystyle \frac{{\theta }_{m_n}}{{\gamma }_{m_n}}}\parallel x_{m_n}-x_{m_n-1}\parallel \hfill \\ & & -2{\beta }_{m_n}\langle w_{m_n}-t_{m_n},x_{m_n+1}-\tilde{p}\rangle +2\langle x_{m_n+1}-\tilde{p},-\tilde{p}\rangle ],\hfill \end{array}$$

(68)

where $M=\underset{n\in \mathbb{N}}{sup}\{\parallel x_{m_n}-\tilde{p}\parallel ,\parallel x_{m_n}-x_{m_n-1}\parallel \}$. This, together with (63), implies that

$$\begin{array}{ccc}\hfill \parallel x_{m_n+1}-\tilde{p}{\parallel }^2& \le & (1-{\gamma }_{m_n})\parallel x_{m_n+1}-\tilde{p}{\parallel }^2+{\gamma }_{m_n}[3M(1-{\gamma }_{m_n}){\displaystyle \frac{{\theta }_{m_n}}{{\gamma }_{m_n}}}\parallel x_{m_n}-x_{m_n-1}\parallel \hfill \\ & & -2{\beta }_{m_n}\langle w_{m_n}-t_{m_n},x_{m_n+1}-\tilde{p}\rangle +2\langle x_{m_n+1}-\tilde{p},-\tilde{p}\rangle ].\hfill \end{array}$$

(69)

Using this one together with (63) again, we obtain

$$\begin{array}{ccc}\hfill \parallel x_n-\tilde{p}{\parallel }^2& \le & 3M(1-{\gamma }_{m_n}){\displaystyle \frac{{\theta }_{m_n}}{{\gamma }_{m_n}}}\parallel x_{m_n}-x_{m_n-1}\parallel -2{\beta }_{m_n}\langle w_{m_n}-t_{m_n},x_{m_n+1}-\tilde{p}\rangle \hfill \\ & & +2\langle x_{m_n+1}-\tilde{p},-\tilde{p}\rangle .\hfill \end{array}$$

(70)

Thus, by using (43), (65), and (67), we have

$$\underset{n\to \infty }{lim\; sup}{\parallel x_n-\tilde{p}\parallel }^2\le 0.$$

Hence, we can conclude that the sequence $\left\{x_n\right\}$ converges strongly to $\tilde{p}:=P_{\mathsf{\Omega }}\left(0\right)$. This completes the proof. □

4. Numerical Experiments

This section will show some examples and numerical results to support Theorems 1 and 2. We will compare the introduced algorithms, IEM and MIEM, with the PPA Algorithm (7) in Example 1 and the PEA Algorithm (8) in Example 2. The numerical experiments are written in Matlab R2015b and performed on a laptop with AMD Dual Core R3-2200U CPU @ 2.50 GHz and RAM 4.00 GB. In both Examples 1 and 2, for each considered matrix, the $\parallel ·\parallel$ means the spectral norm.

Example 1.

Let $H_1={\mathbb{R}}^n$ and $H_2={\mathbb{R}}^m$ be two real Hilbert spaces with the Euclidean norm. We consider the bifunctions $\tilde{f}$ and $\tilde{g}$ which are generated from the Nash–Cournot oligopolistic equilibrium models of electricity markets (see [22,32]),

$$\begin{array}{ccc}\hfill & & \tilde{f}(x,y)=\langle P_{1}x+Q_{1}y,y-x\rangle ,\forall x,y\in {\mathbb{R}}^n,\hfill \end{array}$$

(71)

$$\begin{array}{ccc}\hfill & & \tilde{g}(u,v)=\langle P_{2}u+Q_{2}v,v-u\rangle ,\forall u,v\in {\mathbb{R}}^m,\hfill \end{array}$$

(72)

where $P_1$, $Q_1\in {\mathbb{R}}^{n\times n}$ and $P_2$, $Q_2\in {\mathbb{R}}^{m\times m}$ are matrices such that $Q_1$, $Q_2$ are symmetric positive semidefinite and $Q_1-P_1$, $Q_2-P_2$ are negative semidefinite. Observe that $\tilde{f}(x,y)+\tilde{f}(y,x)={(x-y)}^T(Q_1-P_1)(x-y),\forall x,y\in {\mathbb{R}}^n$. Moreover, from the property of $Q_1-P_1$, we have that $\tilde{f}$ is a monotone operator. Similarly, we also have that $\tilde{g}$ is monotone.

Next, we consider the two bifunctions f and g, which are given by

$$f(x,y)=\left\{\begin{array}{c}\tilde{f}(x,y),if(x,y)\in C\times C,\hfill \\ 0,otherwise,\hfill \end{array}\right.$$

(73)

and

$$g(u,v)=\left\{\begin{array}{c}\tilde{g}(u,v),if(u,v)\in Q\times Q,\hfill \\ 0,otherwise,\hfill \end{array}\right.$$

(74)

where $C={\prod }_{i=1}^n[-5,5]$ and $Q={\prod }_{j=1}^m[-20,20]$ are the constrained boxes. We note that f and g are Lipschitz-type continuous with constants $c_1=c_2=\frac{1}{2}\parallel P_1-Q_1\parallel$ and $d_1=d_2=\frac{1}{2}\parallel P_2-Q_2\parallel$, respectively (see [6]). Choose $b_1=max\{c_1,d_1\}$, and $b_2=max\{c_2,d_2\}$. Then, both bifunctions f and g are Lipschitz-type continuous with constants $b_1$ and $b_2$.

For this numerical experiment, the matrices $P_1$, $Q_1$, $P_2$, and $Q_2$ are randomly generated in the interval $[-5,5]$ such that they satisfy the required properties above and the linear operator $A:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is a $m\times n$ matrix, in which each of its entries is randomly generated in the interval $[-2,2]$. Note that the solution set Ω is nonempty because of $0\in \mathsf{\Omega }$. We will work with the following control parameters: $\alpha =0.5$, $\eta =\frac{1}{{2\parallel A\parallel }^2}$, ${\lambda }_k={\mu }_k=\frac{1}{4max\{b_1,b_2\}}$, ${ϵ}_k=\frac{1}{{(k+1)}^2}$, ${\gamma }_k=\frac{1}{k+1}$, and ${\beta }_k=0.5(1-{\gamma }_k)$. The following five cases of the parameter ${\theta }_k$ are considered:

Case 1. ${\theta }_k=0$.

Case 2. ${\theta }_k=0.25{\overline{\theta }}_k$.

Case 3. ${\theta }_k=0.5{\overline{\theta }}_k$.

Case 4. ${\theta }_k=0.75{\overline{\theta }}_k$.

Case 5. ${\theta }_k={\overline{\theta }}_k$.

The function $quadprog$ in Matlab Optimization Toolbox was used to solve vectors $y_k$, $z_k$, $u_k$, and $v_k$. We randomly generated starting points $x_0=x_1\in {\mathbb{R}}^n$ in the interval $[-5,5]$. The IEM and MIEM algorithms were tested along with the PPA Algorithm (7) by using the stopping criteria $\parallel x_{k+1}-x_k\parallel <{10}^{-8}$. We randomly generated 10 starting points and the presented results are the average, where $n=10$ and $m=20$.

From

Table 1. The numerical results of Example 1.

	Average CPU Times (s)				Average Iterations
Cases	IEM	MIEM	PPA		IEM	MIEM	PPA
1	3.1609	4.1125	1.7781		179.6	237.5	177.6
2	2.7578	3.4859			164.6	207.9
3	2.5641	2.9828			148.7	177.9
4	2.3875	2.4531			131.8	146.7
5	2.0031	1.8766			110.9	112.6

, we may suggest that the parameter ${\theta }_k={\overline{\theta }}_k$ provides better CPU times and iteration numbers than other cases. Moreover, iteration numbers of the IEM and MIEM algorithms with the parameter, ${\theta }_k\ne 0$, are mostly better than those of the PPA Algorithm. Meanwhile, CPU times of the PPA Algorithm are better than those of the IEM and MIEM Algorithms. However, we would like to remind that, by Remark 1, the class of pseudomonotone bifunction is strictly larger than the class of monotone bifunction, and both IEM and MIEM Algorithms can solve the split equilibrium problems for pseudomonotone bifunctions while the PPA Algorithm may not be applied.

Example 2.

Let $H_1={\mathbb{R}}^n$ and $H_2={\mathbb{R}}^m$ be two real Hilbert spaces with the Euclidean norm. We consider a classical form of the bifunction which given by the Cournot–Nash models (see [33]),

$$\begin{array}{ccc}\hfill & & \tilde{f}(x,y)=\langle A_{1}x+a_1^n(y+x),y-x\rangle ,\forall x,y\in {\mathbb{R}}^n,\hfill \end{array}$$

(75)

$$\begin{array}{ccc}\hfill & & \tilde{g}(u,v)=\langle A_{2}u+a_2^m(v+u),v-u\rangle ,\forall u,v\in {\mathbb{R}}^m,\hfill \end{array}$$

(76)

where

$A_1={\left(\begin{array}{ccccc}0& a_1& a_1& \cdots & a_1\\ a_1& 0& a_1& \cdots & a_1\\ a_1& a_1& 0& \cdots & a_1\\ & & & \cdots & \\ a_1& a_1& & \cdots & 0\end{array}\right)}_{n\times n}$, $A_2={\left(\begin{array}{ccccc}0& a_2& a_2& \cdots & a_2\\ a_2& 0& a_2& \cdots & a_2\\ a_2& a_2& 0& \cdots & a_2\\ & & & \cdots & \\ a_2& a_2& & \cdots & 0\end{array}\right)}_{m\times m}$,

where $a_1$ and $a_2$ are positive real numbers. We know that the bifunctions $\tilde{f}$ and $\tilde{g}$ are pseudomonotone and they are not monotone on C and Q, respectively (see [34]).

Here, the numerical experiment is considered under the following setting: the boxes C and Q, the bifunctions f and g, the linear operator A, and the control parameters are given as in Example 1. Notice that the solution set Ω is nonempty because of $0\in \mathsf{\Omega }$. We observe that f and g are Lipschitz-type continuous with constants $c_1=c_2=\frac{1}{2}\parallel A_1\parallel$ and $d_1=d_2=\frac{1}{2}\parallel A_2\parallel$, respectively. Choose $b_1=max\{c_1,d_1\}$, and $b_2=max\{c_2,d_2\}$. Then, both bifunctions f and g are Lipschitz-type continuous with constants $b_1$ and $b_2$. In addition, the positive real numbers $a_1$ and $a_2$ are randomly generated in the interval $(1,2)$ and $(3,4)$, respectively. The following four cases of the parameter ${\theta }_k$ are considered:

Case 1. ${\theta }_k=0.25{\overline{\theta }}_k$.

Case 2. ${\theta }_k=0.5{\overline{\theta }}_k$.

Case 3. ${\theta }_k=0.75{\overline{\theta }}_k$.

Case 4. ${\theta }_k={\overline{\theta }}_k$.

The function $quadprog$ in Matlab Optimization Toolbox was used to solve vectors $y_k$, $z_k$, $u_k$, and $v_k$. The starting points $x_0=x_1\in {\mathbb{R}}^n$ are randomly generated in the interval $[-5,5]$. We compare the IEM and MIEM Algorithms with the PEA Algorithm (8) by using the stopping criteria $\parallel x_{k+1}-x_k\parallel <{10}^{-8}$. We randomly 10 starting points and the presented results are the average, where $n=10$ and $m=20$.

Table 2. The numerical results of Example 2.

		Average CPU Times (s)				Average Iterations
Cases		IEM	MIEM	PPA		IEM	MIEM	PPA
1		1.9609	2.5922	2.1891		79.7	111.4	88.2
2		1.7250	2.2891			71.5	94.6
3		1.5281	1.8844			64.4	78.4
4		1.4094	1.5313			59.2	63.6

shows that the parameter ${\theta }_k$ from case 4, as ${\theta }_k={\overline{\theta }}_k$, yields better CPU times and iteration numbers than other cases. In addition, we see that CPU times and iteration numbers of the IEM and MIEM Algorithms are mostly better than those of the PEA Algorithm.

5. Conclusions

We present two algorithms for solving the split equilibrium problems, when the bifunctions are pseudomonotone and Lipschitz-type continuous in the framework of real Hilbert spaces. We consider both inertial and extragradient methods for introducing a sequence which is convergent to a solution of the considered problem. Some numerical experiments in which the bifunctions are generated from the Nash–Cournot oligopolistic equilibrium models of electricity markets and the Cournot–Nash models, respectively, are performed to illustrate the convergence of introduced algorithms and compare them with some algorithms. In the numerical experiment Example 1, one may observe that the time per iteration ratio of IEM and MIEM is around 0.018, while it is around 0.011 for PPA. This means both the IEM and MIEM Algorithms take approximately 1.63 times the CPU time of the PPA Algorithm in each iteration for solving the considered monotone type problem. When we consider the pseudomonotone type problem in Example 2, the time per iteration of IEM, MIEM, and PEA are about the same at 0.024. This information may lead to a conclusion that a key advantage of the inertial technique is trying to reduce the number of iterations rather than reduce the CPU time per iteration. On the other hand, we emphasize that the exact Lipschitz-type constants of the bifunctions are needed in order to control input parameters of the two introduced algorithms. However, the Lipschitz-type constants of the bifunctions are often unknown, and even in nonlinear problems they are difficult to approximate. For a future research direction, it would be very interesting to develop the algorithm but without the prior knowledge of the Lipschitz-type constants of the bifunctions.