Strong Convergent Theorems Governed by Pseudo-Monotone Mappings

Abstract

The purpose of this paper is to introduce two different kinds of iterative algorithms with inertial effects for solving variational inequalities. The iterative processes are based on the extragradient method, the Mann-type method and the viscosity method. Convergence theorems of strong convergence are established in Hilbert spaces under mild assumption that the associated mapping is Lipschitz continuous, pseudo-monotone and sequentially weakly continuous. Numerical experiments are performed to illustrate the behaviors of our proposed methods, as well as comparing them with the existing one in literature.

1. Introduction and Preliminaries

Let C be a closed, convex and nonempty subset of a Hilbert space H. The inner product and its induced norm of H are denoted by $\langle ·,·\rangle$ and $\parallel ·\parallel$ respectively.

Problem 1.

Let $\mathcal{A}:\mathcal{H}\to \mathcal{H}$ be a nonlinear operator. We consider the following variational inequality problem

$$\mathit{find}{x}^{*}\in VI(C,A):=\{x^{*}\in C:\langle z-x^{*},A\left(x^{*}\right)\rangle \ge 0,\forall z\in C\}.$$

(1)

The variational inequality, which serves as an important model in studying a wide class of real problems arising in traffic network, medical imaging, machine learning, transportation, etc. Due to its wide applications, this model unifies a number of optimization-related problems, such as, saddle problems, equilibrium problems, complementary problems, fixed point problems; see, e.g., [1,2,3,4,5].

Next, one introduces an important tool in this paper: the nearest point (metric) projection. For all $x\in H$, there exists a unique nearest point in C, which is denoted by $P_{C}x$, such that $\parallel x-P_{C}x\parallel =inf\{\parallel x-y\parallel :y\in C\}.$ Then $P_C$ is called the nearest point (metric) projection from H onto C. It is known that the projection operator is firmly nonexpansive and can be characterized by $\langle x-P_{C}x,y-P_{C}x\rangle \le 0,$$\forall x\in H,y\in C$, which is also equivalent to $\parallel x-P_C{x\parallel }^2\le {\parallel x-y\parallel }^2-{\parallel y-P_{C}x\parallel }^2,$$\forall x\in H,y\in C$; see [6]. As a routine way, one can turn the variational inequality problem into a fixed point problem via a resolvent operator, that is, $P_C(I-\alpha A)x^{*}=x^{*},$ for all $\alpha >0.$

Let us recall some definitions of mappings involved in our study. An operator $A:H\to H$ is said to be

(i)

Sequentially weakly continuous if, for each weak sequence $\left\{x_k\right\}$ to $x^{*}$, in this case we say that $\left\{A{x}_k\right\}$ converges weakly to $A{x}^{*}$;

(ii)

Pseudomonotone on H if $\langle Ax,y-x\rangle \ge 0\Rightarrow \langle Ay,y-x\rangle \ge 0,$$\forall x,y\in H;$

(iii)

Monotone on H if $\langle Ax-Ay,x-y\rangle \ge 0,$$\forall x,y\in H;$

(iv)

L-Lipschitz continuous on H if there exists $L>0$ such that $\parallel Ax-Ay\parallel \le L\parallel x-y\parallel ,$$\forall x,y\in H$.

Suppose that $g:H\to \mathbb{R}$ is convex and continuously Fréchet differentiable. Then g is convex if and only if $\nabla g:H\to H$ is monotone. At the same time, it is well known that $\nabla g$ is pseudo-monotone if and only if g is pseudo-convex.

Recently, iterative methods for solving variational inequalities and related optimization problems have been proposed and analyzed by many authors [7,8,9,10,11]. Let us start with Korpelevich’s method [12], which was proposed for the Euclidean case, known as the extragradient method. This method needs two calculations of the projection onto a nonempty closed convex subset, that is, it generates a sequence by the following iteration procedure

$$\left\{\begin{array}{c}{x}_0\in C,\hfill \\ w_k=P_C(x_k-\lambda A{x}_k),\hfill \\ x_{k+1}=P_C(x_k-\lambda A{w}_k),\hfill \end{array}\right.$$

(2)

where the mapping $A:H\to H$ is monotone and L-Lipschitz continuous for some $L>0$ and $\lambda L\in (0,1)$. Recently, the extragradient method has received great attention by many authors [13]. It has been studied in various ways for solving a more general problem in the setting of Hilbert spaces. Typically, the extragradient method has been successfully applied for solving the pseudomonotone variational inequality, see [14].

Now, let us consider the inertial extrapolation, which can be regarded as an acceleration procedure of speeding up the convergence properties. Due to its importance, there are increasing interests in studying inertial-type algorithms; see, e.g., [15,16,17,18,19] and the references therein. By incorporating the inertial extrapolation into the extragradient method, Dong et al. [20] introduced the following inertial extragradient algorithm (EAI). Given any $x_0,x_1\in H$,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{z}_k=x_k+{\gamma }_k(x_k-x_{k-1}),\hfill \\ y_k=P_C(z_k-\beta A{z}_k),\hfill \\ x_{k+1}=(1-{\alpha }_k)z_k+{\alpha }_k{P}_C(z_k-\beta A{y}_k),\hfill \end{array}\right.\end{array}$$

(3)

for each $k\ge 1$, where the mapping $A:H\to H$ is monotone and Lipschitz continuous with the constant $L>0$. The authors showed that $\left\{x_k\right\}$ converges weakly to an element of VI($C,A$) under the following conditions:

(i)

$\left\{{\gamma }_k\right\}$ is non-decreasing with ${\gamma }_1=0$ and $0\le {\gamma }_k\le \gamma <1$ for each $k\ge 1$;

(ii)

$\exists \alpha ,\sigma ,\delta >0$ such that

$$\begin{array}{c}\hfill \frac{\gamma [{(1+\beta L)}^2\gamma (1+\gamma )+(1-{\beta }^2{L}^2)\gamma \sigma +\sigma {(1+\beta L)}^2]}{1-{\beta }^2{L}^2}<\delta \end{array}$$

and

$$\begin{array}{c}\hfill \frac{(1-{\beta }^2{L}^2)\delta -\gamma [{(1+\beta L)}^2\gamma (1+\gamma )+(1-{\beta }^2{L}^2)\gamma \sigma +\sigma {(1+\beta L)}^2]}{\delta [{(1+\beta L)}^2\gamma (1+\gamma )+(1-{\beta }^2{L}^2)\gamma \sigma +\sigma {(1+\beta L)}^2]}\ge {\alpha }_k\ge \alpha >0.\end{array}$$

Note that inertial extragradient algorithm involving the inertial term mentioned above is weakly convergent.

Many problems, arising in a broad range of applied areas, such as image recovery, quantum physics, economics, control theory and mechanics, have been extensively studied in the infinite dimensional setting. In such problems, the norm convergence is essential, since the energy $\parallel x-x_n{\parallel }^2$ of the error between the solution x and the iterate $x_n$ eventually becomes arbitrarily small. Furthermore, in the context of solving the convex optimization problem $min\{\Lambda (x):x\in H\},$ the rate of convergence of $\{\Lambda \left(x_k\right)\}$ seems to be better in the case when $\left\{x_k\right\}$ enjoys a strong convergence than that in the case when $\left\{x_k\right\}$ enjoys a weak convergence. Thus, these naturally give rise to a question how to appropriately modify the inertial extragradient method so that the strong convergence is guaranteed. To solve the above question, we will propose two modified inertial extragradient algorithms. The first modification stems from the Mann-type method [21,22] and the other one is of viscosity nature [23].

An obvious disadvantage of Algorithms (2) and (3) is the assumption that the mapping A should be Lipschitz continuous and monotone. To avoid this restrictive assumption, in this paper, we show that our proposed algorithms can solve the pseudo-monotone variational inequality under suitable assumptions. It is worth mentioning that the class of pseudo-monotone mappings virtually contains the class of monotone mappings. Indeed, the scope of the related optimization problems can be enlarged from convex optimization problems to pseudoconvex optimization problems. This guarantees the advantage of modified inertial extragradient methods in comparing with the other solution methods.

The following lemmas will be used in the proof of our main results.

Lemma 1

([24]). Let $\left\{a_k\right\}$ be nonnegative real sequence with relation $a_{k+1}\le {\alpha }_k{b}_k+a_k(1-{\alpha }_k),$ where $\left\{{\alpha }_k\right\}\subset (0,1)$ and $\left\{b_k\right\}$ are real sequences satisfying the restrictions ${lim}_{k\to \infty }{\alpha }_k=0$, ${\sum }_{k=0}^{\infty }{\alpha }_k=\infty$ and ${lim\; sup}_{k\to \infty }{b}_k\le 0$. Then $a_k\to 0$ as $k\to \infty .$

Lemma 2

([25]). Let $\left\{a_k\right\}$ be a real sequence defined in $[0,+\infty )$ such that there exists a subsequence $\left\{a_{kj}\right\}$ of $\left\{a_k\right\}$ with $a_{k_j}<a_{k_j+1}$ for all $j\in \mathbb{N}$. There exists a nondecreasing sequence $\left\{m_i\right\}$ such that ${lim}_i{m}_i=\infty$ and the following properties are satisfied by all (sufficiently large) number $i\in \mathbb{N}$: $a_{m_i+1}\ge a_i$ and $a_{m_i+1}\ge a_{m_i}$. Indeed, $m_i$ is the largest number of n in the range of $\{1,2,\dots ,i\}$ such that $a_k\le a_{k+1}$ holds.

The rest of the paper is organized as follows. In Section 2, we give two variants of the inertial extragradient method for solving pseudo-monotone variational inequalities. We also prove strong convergence results for the proposed algorithms. In Section 3, some numerical experiments are presented to deal with quadratic programming problems, which demonstrates the performances of our methods. Finally, the conclusion is given in the last section.

2. Algorithm and Convergence

Throughout the rest of the paper, one always systematically assumes that the following set of hypotheses:

• The feasible set C is a nonempty, closed and convex set in a real Hilbert space H;
• The operator $A:H\to H$ is pseudo-monotone, sequentially weakly continuous and L-Lipschitz continuous for some $L>0$, with its solution set $VI(C,A)\ne \varnothing$.

First, we present the algorithm for solving the pseudo-monotone variational inequality which combines the inertial extragradient method and Mann-type method.

The following propositions are known results of the iterative sequences generated by Algorithms 1 and 2, which are crucial for the proof of our convergence theorems, see [14].

Proposition 1.

Assume that A is pseudo-monotone and L-Lipschitz continuous with $VI(C,A)\ne \varnothing$. Let $x^{*}$ be a solution of $VI(C,A)$. Setting $z_k=P_C(y_k-{\lambda }_{k}A{y}_k)$, we have

$$\parallel z_k-x^{*}{\parallel }^2\le \parallel w_k-x^{*}{\parallel }^2-(1-{\lambda }_k^2{L}^2){\parallel y_k-w_k\parallel }^2.$$

Proposition 2.

The mapping A is pseudo-monotone, sequentially weakly continuous and L-Lipschitz continuous for some $L>0$. Assume that $VI(C,A)\ne \varnothing$. Assume additionally that ${lim}_{k\to \infty }\parallel y_k-w_k\parallel =0$ and ${lim\; inf}_{k\to \infty }{\lambda }_k>0$. Then the sequence $\left\{w_k\right\}$ generated by Algorithms 1 and 2 converges weakly to a solution of $VI(C,A)$.

Algorithm 1:

Now we are in a position to establish the main result of this note.

Theorem 1.

Let $\left\{{\lambda }_k\right\}$, $\left\{{\gamma }_k\right\}$, $\left\{{\beta }_k\right\}$ be three real sequences in $(0,1)$ such that $0<a\le {\lambda }_n\le b<\frac{1}{L}$ for some $a,b\in (0,1)$, ${lim}_{k\to \infty }{\gamma }_k=0$, ${\sum }_{k=1}^{\infty }{\gamma }_k=\infty$, $\left\{{\beta }_k\right\}\subset (0,1-{\gamma }_k)$ and ${lim}_{k\to \infty }{\beta }_k>0$. Assume that the sequence $\left\{{\alpha }_k\right\}$ is chosen such that ${lim}_{k\to \infty }\frac{{\alpha }_k}{{\gamma }_k}\parallel x_k-x_{k-1}\parallel =0$. Then the sequence $\left\{x_n\right\}$ generated by Algorithm 1 converges to the solution $\widehat{x}=P_{VI(C,A)}0$ in norm.

Proof. 

Let us fix $x^{*}=P_{VI(C,A)}0$. To simplify the notations, one sets $z_k=P_C(y_k-{\lambda }_{k}A{y}_k)$. By applying Proposition 1, together with the definition of $\left\{w_k\right\}$, one easily obtains that

$$\begin{array}{c}\hfill \parallel z_k-x^{*}\parallel \le \parallel w_k-x^{*}\parallel =\parallel x_k+{\alpha }_k(x_k-x_{k-1})-x^{*}\parallel \le {\alpha }_k\parallel x_k-x_{k-1}\parallel +\parallel x_k-x^{*}\parallel .\end{array}$$

(4)

Invoking (4), the definition of $\left\{x_k\right\}$ implies that

$$\begin{array}{cc}\hfill \parallel x_{k+1}-x^{*}\parallel \le & \parallel (1-{\beta }_k-{\gamma }_k)(x_k-x^{*})+{\beta }_k(z_k-x^{*})\parallel +{\gamma }_k\parallel x^{*}\parallel \hfill \\ \hfill \le & (1-{\gamma }_k)\parallel x_k-x^{*}\parallel +{\gamma }_k\left(\frac{{\alpha }_k}{{\gamma }_k}\parallel x_k-x_{k-1}\parallel +\parallel x^{*}\parallel \right).\hfill \end{array}$$

(5)

Let $M_1>0$ be a positive constant such that $\frac{{\alpha }_k}{{\gamma }_k}\parallel x_k-x_{k-1}\parallel \le M_1$. Due to the assumption ${lim}_{k\to \infty }\frac{{\alpha }_k}{{\gamma }_k}\parallel x_k-x_{k-1}\parallel =0$. Coming back to (5), we obtain that

$$\begin{array}{c}\hfill \parallel x_{n+1}-x^{*}\parallel \le max\{\parallel x_k-x^{*}\parallel ,M_1+\parallel x^{*}\parallel \}\le \dots \le max\{\parallel x_0-x^{*}\parallel ,M_1+\parallel x^{*}\parallel \}.\end{array}$$

(6)

This clearly implies that $\left\{x_k\right\}$ is bounded. As a result, we have that sequences $\left\{y_k\right\}$, $\left\{w_k\right\}$ and $\left\{z_k\right\}$ are bounded as well. Again, by using the definition of $\left\{x_k\right\}$, we find that

$$\begin{array}{cc}\hfill & \parallel x_{k+1}-x^{*}{\parallel }^2\hfill \\ =\hfill & \parallel {\beta }_k(z_k-x^{*})+(1-{\beta }_k-{\gamma }_k)(x_k-x^{*})-{\gamma }_k{x}^{*}{\parallel }^2\hfill \\ \le \hfill & \parallel {\beta }_k(z_k-x^{*})+(1-{\beta }_k-{\gamma }_k)(x_k-x^{*}){\parallel }^2-2{\gamma }_k\langle (1-{\beta }_k-{\gamma }_k)(x_k-x^{*})+{\beta }_k(z_k-x^{*}),x^{*}\rangle \hfill \\ \le \hfill & \parallel {\beta }_k(z_k-x^{*})+(1-{\beta }_k-{\gamma }_k)(x_k-x^{*}){\parallel }^2+2{\gamma }_k\parallel (1-{\beta }_k-{\gamma }_k)(x_k-x^{*})+{\beta }_k(z_k-x^{*})\parallel \parallel x^{*}\parallel .\hfill \end{array}$$

(7)

On the other hand,

$$\begin{array}{cc}\hfill & \parallel {\beta }_k(z_k-x^{*})+(1-{\beta }_k-{\gamma }_k)(x_k-x^{*}){\parallel }^2\hfill \\ \hfill \le & {\beta }_k^2\parallel z_k-x^{*}{\parallel }^2+{(1-{\beta }_k-{\gamma }_k)}^2\parallel x_k-x^{*}{\parallel }^2+2(1-{\beta }_k-{\gamma }_k){\beta }_k\parallel x_k-x^{*}\parallel \parallel z_k-x^{*}\parallel \hfill \\ \hfill \le & {\beta }_k^2\parallel z_k-x^{*}{\parallel }^2+{(1-{\beta }_k-{\gamma }_k)}^2\parallel x_k-x^{*}{\parallel }^2+(1-{\beta }_k-{\gamma }_k){\beta }_k{\parallel x_k-x^{*}\parallel }^2\hfill \\ \hfill & +(1-{\beta }_k-{\gamma }_k){\beta }_k{\parallel z_k-x^{*}\parallel }^2\hfill \\ \hfill \le & (1-{\gamma }_k){\beta }_k\parallel z_k-x^{*}{\parallel }^2+(1-{\beta }_k-{\gamma }_k)(1-{\gamma }_k){\parallel x_k-x^{*}\parallel }^2.\hfill \end{array}$$

(8)

In view of the definition of $\left\{w_k\right\}$, we deduce that

$$\begin{array}{cc}\hfill \parallel w_k-x^{*}{\parallel }^2\le & \parallel {\alpha }_k(x_k-x_{k-1})+x_k-x^{*}{\parallel }^2\hfill \\ \hfill \le & 2{\alpha }_k\parallel x_k-x_{k-1}\parallel \parallel w_k-x^{*}\parallel +\parallel x_k-x^{*}{\parallel }^2.\hfill \end{array}$$

(9)

Invoking the boundedness of $\left\{x_k\right\}$ and $\left\{z_k\right\}$, there exists a positive constant $M_2>0$ such that

$$\parallel (1-{\beta }_k-{\gamma }_k)(x_k-x^{*})+{\beta }_k(z_k-x^{*})\parallel \parallel x^{*}\parallel \le M_2.$$

(10)

By combining inequalities (7)–(10) with Proposition 1, one asserts that

$$\begin{array}{cc}\hfill & \parallel x_{n+1}-x^{*}{\parallel }^2\hfill \\ \hfill \le & (1-{\beta }_k-{\gamma }_k)(1-{\gamma }_k)\parallel x_k-x^{*}{\parallel }^2+(1-{\gamma }_k){\beta }_k{\parallel z_k-x^{*}\parallel }^2+2{\gamma }_k{M}_2\hfill \\ \hfill \le & (1-{\beta }_k-{\gamma }_k)(1-{\gamma }_k)\parallel x_k-x^{*}{\parallel }^2+(1-{\gamma }_k){\beta }_k(\parallel x_k-x^{*}{\parallel }^2+2{\alpha }_k\parallel x_k-x_{k-1}\parallel \parallel w_k-x^{*}\parallel \hfill \\ \hfill & -(1-{\lambda }_k^2{L}^2)\parallel y_k-w_k{\parallel }^2)+2{\gamma }_k{M}_2\hfill \\ \hfill \le & \parallel x_k-x^{*}{\parallel }^2+2{\alpha }_k\parallel x_k-x_{k-1}\parallel \parallel w_k-x^{*}\parallel -2(1-{\gamma }_k){\beta }_k(1-{\lambda }_k^2{L}^2){\parallel y_k-w_k\parallel }^2+2{\gamma }_k{M}_2.\hfill \end{array}$$

(11)

Note that $x_{k+1}=(1-{\beta }_k)x_k+{\beta }_k{z}_k-{\gamma }_k{x}_k.$ Setting $p_n=(1-{\beta }_k)x_k+{\beta }_k{z}_k$, we find that $p_k-x_k={\beta }_k(z_k-x_k)$. Furthermore, we can reformulate $x_{k+1}$ as

$$x_{k+1}=p_k-{\gamma }_k{x}_k=(1-{\gamma }_k)p_k+{\gamma }_k(p_k-x_k)=(1-{\gamma }_k)p_k+{\gamma }_k{\beta }_k(z_k-x_k).$$

It follows from the above equality that

$$\begin{array}{cc}\hfill & \parallel x_{k+1}-x^{*}{\parallel }^2\hfill \\ \hfill =& \parallel (1-{\gamma }_k)(p_k-x^{*})+{\gamma }_k({\beta }_k(z_k-x_k)-x^{*}){\parallel }^2\hfill \\ \hfill \le & (1-{\gamma }_k){\parallel p_k-x^{*}\parallel }^2+2{\gamma }_k{\beta }_k\langle z_k-x_k,x_{k+1}-x^{*}\rangle -2{\gamma }_k\langle x^{*},x_{k+1}-x^{*}\rangle .\hfill \end{array}$$

(12)

Indeed, based on (9), we have that

$$\begin{array}{cc}\hfill \parallel p_k-x^{*}{\parallel }^2=& \parallel {\beta }_k(z_k-x^{*})+(1-{\beta }_k)(x_k-x^{*}){\parallel }^2\hfill \\ \hfill \le & (1-{\beta }_k)\parallel x_k-x^{*}{\parallel }^2+{\beta }_k(2{\alpha }_k\parallel x_k-x_{k-1}\parallel \parallel w_k-x^{*}\parallel +\parallel x_k-x^{*}{\parallel }^2)\hfill \\ \hfill \le & 2{\alpha }_k\parallel x_k-x_{k-1}\parallel \parallel w_k-x^{*}\parallel +\parallel x_k-x^{*}{\parallel }^2.\hfill \end{array}$$

(13)

Combining (12) with (13), we find that

$$\begin{array}{cc}\hfill \parallel x_{k+1}-x^{*}{\parallel }^2\le & (1-{\gamma }_k)(\parallel x_k-x^{*}{\parallel }^2+2{\alpha }_k\parallel x_k-x_{k-1}\parallel \parallel w_k-x^{*}\parallel )\hfill \\ \hfill & +2{\gamma }_k{\beta }_k\parallel z_k-x_k\parallel \parallel x_{k+1}-x^{*}\parallel -2{\gamma }_k\langle x^{*},x_{k+1}-x^{*}\rangle \hfill \\ \hfill \le & (1-{\gamma }_k)\parallel x_k-x^{*}{\parallel }^2+{\gamma }_k(\frac{2{\alpha }_k}{{\gamma }_k}\parallel x_k-x_{k-1}\parallel \parallel w_k-x^{*}\parallel \hfill \\ \hfill & +2{\beta }_k\parallel z_k-x_k\parallel \parallel x_{k+1}-x^{*}\parallel +2\langle x^{*},x^{*}-x_{k+1}\rangle ).\hfill \end{array}$$

(14)

Now we prove that the sequence $\{\parallel x_k-x^{*}\parallel \}$ converges to 0 by considering two possible cases on $\{\parallel x_k-x^{*}\parallel \}$.

Case 1. Suppose that there exists $K\in \mathbb{N}$ such that $\parallel x_{k+1}-x^{*}\parallel \le \parallel x_k-x^{*}\parallel$ for all $k\ge K$. This implies that ${lim}_{k\to \infty }\parallel x_k-x^{*}\parallel$ exists. From conditions ${lim}_{k\to \infty }\frac{{\alpha }_k}{{\gamma }_k}\parallel x_k-x_{k-1}\parallel =0$ and $\left\{{\gamma }_k\right\}\in (0,1)$, we find that ${lim}_{k\to \infty }{\alpha }_k\parallel x_k-x_{k-1}\parallel =0$. Since $0<a\le {\lambda }_n\le b<\frac{1}{L}$, it holds that

$$0<1-b^2{L}^2\le 1-{\lambda }_k^2{L}^2\le 1-a^2{L}^2<1.$$

(15)

It follows from (11) and (15) and the conditions ${lim}_{k\to \infty }{\gamma }_k=0$ and ${lim}_{k\to \infty }{\beta }_k>0$ that

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

(16)

From the nonexpansivity of $P_C$ and the L-Lipschitz continuity of A, one concludes that

$$\parallel z_k-y_k\parallel =\parallel P_C(y_k-{\lambda }_{k}A{y}_k)-P_C(w_k-{\lambda }_{k}A{w}_k)\parallel \le (1+{\lambda }_{k}L)\parallel y_k-w_k\parallel .$$

(17)

Combining (16) with (17), one has

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

(18)

It follows that

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

(19)

From (16), (18) and (19), we obtain that

$$\underset{k\to \infty }{lim}\parallel z_k-x_k\parallel \le \underset{k\to \infty }{lim}(\parallel z_k-y_k\parallel +\parallel y_k-w_k\parallel +\parallel w_k-x_k\parallel )=0.$$

(20)

Recalling that $\left\{x_k\right\}$ is bounded, one assumes that there exists a subsequence $\left\{x_{k_j}\right\}$ of $\left\{x_k\right\}$ such that $x_{k_j}\rightharpoonup \widehat{x}$ as $j\to \infty$. Invoking (19), one has that $w_{k_j}\rightharpoonup \widehat{x}$ as $j\to \infty$. As a sequence, by use of Proposition 2, we find that $\widehat{x}\in VI(C,A)$. From the fact $x^{*}=P_{VI(C,A)}0$, we obtain that

$$\underset{k\to \infty }{lim\; sup}\langle x^{*},x^{*}-x_k\rangle =\underset{j\to \infty }{lim}\langle x^{*},x^{*}-x_{k_j}\rangle =\langle x^{*},x^{*}-\widehat{x}\rangle \le 0.$$

(21)

From the boundedness of $\left\{x_k\right\}$ and ${lim}_{k\to \infty }{\gamma }_k=0$, we infer

$$\parallel x_{k+1}-x_k\parallel =\parallel {\beta }_k(z_k-x_k)-{\gamma }_k{x}_k\parallel \le {\beta }_k\parallel z_k-x_k\parallel +{\gamma }_k\parallel x_k\parallel \to 0,k\to \infty .$$

(22)

Combining (21) with (22), we further find that

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

(23)

From the condition ${lim}_{k\to \infty }\frac{{\alpha }_k}{{\gamma }_k}\parallel x_k-x_{k+1}\parallel ={lim}_{k\to \infty }{\gamma }_k=0$, ${\sum }_{k=1}^{\infty }{\gamma }_k=\infty$, (14), (20) and (23), we conclude from Lemma 1 that ${lim}_{k\to \infty }\parallel x_k-x^{*}\parallel =0$. In other words, it entails that $x_k\to x^{*}$ as $k\to \infty$.

Case 2: Suppose that there exists a subsequence $\{\parallel x_{k_j}-p\parallel \}$ of $\{\parallel x_k-p\parallel \}$ such that $\parallel x_{k_j}-p\parallel <\parallel x_{k_j+1}-p\parallel ,$ $\forall j\in \mathbb{N}$. In this case, by using Lemma 2, one sees that there exists a nondecreasing sequence $\left\{n_i\right\}$ of $\mathbb{N}$ such that ${lim}_{i\to \infty }{n}_i=\infty$ and the following inequalities hold for all $i\in \mathbb{N}$,

$$\parallel x_{n_i+1}-p\parallel \ge \parallel x_{n_i}-p\parallel ,\parallel x_{n_i+1}-p\parallel \ge \parallel x_i-p\parallel .$$

(24)

By using (11) and (24), we have that

$$\begin{array}{c}\hfill 2(1-{\gamma }_{n_i}){\beta }_{n_i}(1-{\lambda }_{n_i}^2{L}^2)\parallel y_{n_i}-w_{n_i}{\parallel }^2\le 2{\alpha }_{n_i}\parallel x_{n_i}-x_{n_i-1}\parallel \parallel w_{n_i}-x^{*}\parallel +2{\gamma }_{n_i}{M}_2.\end{array}$$

(25)

Recalling that ${lim}_{k\to \infty }{\alpha }_{n_i}\parallel x_{n_i}-x_{n_i-1}\parallel ={lim}_{k\to \infty }{\gamma }_{n_i}=0$ and ${lim}_{k\to \infty }{\beta }_{n_i}>0$, it follows from (15) and (25) that

$$\underset{i\to \infty }{lim}\parallel y_{n_i}-w_{n_i}\parallel =0.$$

(26)

Using the same arguments as in the proof of Case 1, one obtains that

$$\underset{i\to \infty }{lim}\parallel w_{n_i}-x_{n_i}\parallel =\underset{i\to \infty }{lim}\parallel z_{n_i}-x_{n_i}\parallel =\underset{i\to \infty }{lim}\parallel x_{n_i+1}-x_{n_i}\parallel =0.$$

(27)

and

$$\underset{i\to \infty }{lim\; sup}\langle x^{*},x^{*}-x_{n_i+1}\rangle \le 0.$$

(28)

Coming back to (14), we have

$$\begin{array}{c}\hfill \parallel x_{n_i+1}-x^{*}{\parallel }^2\le \frac{2{\alpha }_{n_i}}{{\gamma }_{n_i}}\parallel x_{n_i}-x_{n_i-1}\parallel \parallel w_{n_i}-x^{*}\parallel +2{\beta }_{n_i}\parallel z_{n_i}-x_{n_i}\parallel \parallel x_{n_i+1}-x^{*}\parallel +2\langle x^{*},x^{*}-x_{n_i+1}\rangle .\end{array}$$

(29)

In light of (27)–(29), we have ${lim\; sup}_{i\to \infty }{\parallel x_{n_i+1}-x^{*}\parallel }^2\le 0$. Invoking (24), we obtain that ${lim}_{i\to \infty }{\parallel x_i-x^{*}\parallel }^2=0$, which further implies that $x_k\to x^{*}$, as $k\to \infty$. This completes the proof.  □

The other algorithm reads as follows.

Algorithm 2:

Now, we are ready to analyze the convergence of Algorithm 2. The outline of its proof is similar to that of Theorem 1.

Theorem 2.

Let $g:H\to H$ be a contraction mapping with the contraction parameter $\kappa \in (0,1)$. Let $\left\{{\lambda }_k\right\}$, $\left\{{\delta }_k\right\}$ be two real sequences in $(0,1)$ such that $0<a\le {\lambda }_n\le b<\frac{1}{L}$ for some $a,b\in (0,1)$, ${\sum }_{k=1}^{\infty }{\delta }_k=\infty$ and ${lim}_{k\to \infty }{\delta }_k=0$. Assume that the sequence $\left\{{\alpha }_k\right\}$ is chosen such that ${lim}_{k\to \infty }\frac{{\alpha }_k}{{\delta }_k}\parallel x_k-x_{k-1}\parallel =0$. Then the sequence $\left\{x_n\right\}$ generated by Algorithm 2 converges strongly to the solution $\widehat{x}\in VI(C,A)$, where $\widehat{x}=P_{VI(C,A)}g\left(\widehat{x}\right)$.

Proof. 

Fixing $x^{*}=P_{VI(C,A)}\circ g\left(x^{*}\right)$ and using the same arguments as in the proof of Theorem 1, we infer

$$\parallel z_k-x^{*}\parallel \le \parallel w_k-x^{*}\parallel \le \parallel x_k-x^{*}\parallel +{\alpha }_k\parallel x_k-x_{k-1}\parallel ,$$

(30)

$$\parallel w_k-x^{*}{\parallel }^2\le \parallel x_k-x^{*}{\parallel }^2+2{\alpha }_k\parallel x_k-x_{k-1}\parallel \parallel w_k-x^{*}\parallel ,$$

(31)

and

$$\parallel z_k-y_k\parallel \le (1+{\lambda }_{k}L)\parallel y_k-w_k\parallel .$$

(32)

Now, using (30) and the definition of $\left\{x_k\right\}$, one sees that

$$\begin{array}{cc}\hfill & \parallel x_{k+1}-x^{*}\parallel \hfill \\ \hfill & \le (1-{\delta }_k)\parallel z_k-x^{*}\parallel +{\delta }_k\parallel g\left(z_k\right)-g\left(x^{*}\right)\parallel +{\delta }_k\parallel g\left(x^{*}\right)-x^{*}\parallel \hfill \\ \hfill & \le (1-{\delta }_k(1-\kappa ))\parallel z_k-x^{*}\parallel +{\delta }_k\parallel g\left(x^{*}\right)-x^{*}\parallel \hfill \\ \hfill & \le (1-{\delta }_k(1-\kappa ))\parallel x_k-x^{*}\parallel +{\delta }_k(1-\kappa )\left(\frac{{\alpha }_k}{{\delta }_k(1-\kappa )}\parallel x_k-x_{k-1}\parallel +\frac{1}{1-\kappa }\parallel g\left(x^{*}\right)-x^{*}\parallel \right).\hfill \end{array}$$

(33)

Since ${lim}_{k\to \infty }\frac{{\alpha }_k}{{\delta }_k}\parallel x_k-x_{k-1}\parallel =0$, we set briefly $\frac{{\alpha }_k}{{\delta }_k}\parallel x_k-x_{k-1}\parallel \le M_3$ for some positive constant $M_3>0$. Coming back to (33), we have that

$$\begin{array}{cc}\hfill \parallel x_k-x^{*}\parallel & \le (1-{\delta }_k(1-\kappa ))\parallel x_k-x^{*}\parallel +{\delta }_k(1-\kappa )\left(\frac{M_3}{1-\kappa }+\frac{1}{1-\kappa }(\parallel g\left(x^{*}\right)-x^{*}\parallel )\right)\hfill \\ \hfill & \le max\left\{\parallel x_k-x^{*}\parallel ,\frac{M_3}{1-\kappa }+\frac{1}{1-\kappa }(\parallel g\left(x^{*}\right)-x^{*}\parallel )\right\}\hfill \\ \hfill & \le \cdots \le max\left\{\parallel x_0-x^{*}\parallel ,\frac{M_3}{1-\kappa }+\frac{1}{1-\kappa }(\parallel g\left(x^{*}\right)-x^{*}\parallel )\right\}.\hfill \end{array}$$

It entails that $\left\{x_k\right\}$ is bounded. This implies that $\left\{w_k\right\}$, $\left\{y_k\right\}$ and $\left\{z_k\right\}$ are bounded as well. We apply Proposition 1 and (31) to get that

$$\begin{array}{cc}\hfill \parallel x_{k+1}-x^{*}{\parallel }^2& \le (1-{\delta }_k)\parallel z_k-x^{*}{\parallel }^2+{\delta }_k(\parallel g\left(z_k\right)-g\left(x^{*}\right)\parallel +\parallel g\left(x^{*}\right)-x^{*}{\parallel )}^2\hfill \\ \hfill & \le (1-{\delta }_k)\parallel z_k-x^{*}{\parallel }^2+{\delta }_k(\parallel z_k-x^{*}\parallel +\parallel g\left(x^{*}\right)-x^{*}{\parallel )}^2\hfill \\ \hfill & \le \parallel x_k-x^{*}{\parallel }^2+2{\alpha }_k\parallel x_k-x_{k-1}\parallel \parallel w_k-x^{*}\parallel -(1-{\lambda }_k^2{L}^2){\parallel y_k-w_k\parallel }^2\hfill \\ \hfill & +{\delta }_k(\parallel g\left(x^{*}\right)-x^{*}{\parallel }^2+2\parallel z_k-x^{*}\parallel \parallel g\left(x^{*}\right)-x^{*}\parallel ).\hfill \end{array}$$

(34)

Again, by using Proposition 1 and (31), together with (9), one concludes that

$$\begin{array}{cc}\hfill & \parallel x_{k+1}-x^{*}{\parallel }^2\hfill \\ \hfill & \le (1-{\delta }_k)\parallel z_k-x^{*}{\parallel }^2+{\delta }_k{\parallel g\left(z_k\right)-g\left(x^{*}\right)\parallel }^2+2{\delta }_k\langle g\left(x^{*}\right)-x^{*},x_{k+1}-x^{*}\rangle \hfill \\ \hfill & \le (1-{\delta }_k(1-\kappa )){\parallel z_k-x^{*}\parallel }^2+2{\delta }_k\langle g\left(x^{*}\right)-x^{*},x_{k+1}-x^{*}\rangle \hfill \\ \hfill & \le (1-{\delta }_k(1-\kappa ))\parallel x_k-x^{*}{\parallel }^2+{\delta }_k(1-\kappa )(\frac{2{\alpha }_k}{{\delta }_k(1-\kappa )}\parallel x_k-x_{k-1}\parallel \parallel w_k-x^{*}\parallel \hfill \\ \hfill & +\frac{2}{1-\kappa }\langle g\left(x^{*}\right)-x^{*},x_{k+1}-x^{*}\rangle ).\hfill \end{array}$$

(35)

Now let us show that the sequence $\{\parallel x_k-x^{*}\parallel \}$ converges to zero. To obtain this result, we consider two possible cases on the sequence $\{\parallel x_k-x^{*}\parallel \}$.

Case 1: There exists $K\in \mathbb{N}$ such that $\parallel x_{k+1}-x^{*}\parallel \le \parallel x_k-x^{*}\parallel$ for all $k\ge K$. Observe that ${lim}_{k\to \infty }{\parallel x_k-x^{*}\parallel }^2$ exists. Due to the condition $0<a\le {\lambda }_n\le b<\frac{1}{L}$, we have that

$$0<1-b^2{L}^2\le 1-{\lambda }_k^2{L}^2\le 1-a^2{L}^2<1.$$

(36)

Based on the conditions ${lim}_{k\to \infty }\frac{{\alpha }_k}{{\delta }_k}\parallel x_k-x_{k-1}\parallel =0$ and $\left\{{\delta }_k\right\}\subset (0,1)$, we have that ${lim}_{k\to \infty }{\alpha }_k\parallel x_k-x_{k-1}\parallel =0$. Since $\left\{w_k\right\}$ and $\left\{z_k\right\}$ are bounded and the condition ${lim}_{k\to \infty }{\delta }_k=0$ holds, we find that

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

(37)

Combining (32) with (37), we have that

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

(38)

It follows that

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

(39)

In light of (37)–(39), we have that

$$\underset{k\to \infty }{lim}\parallel z_k-x_k\parallel \le \underset{k\to \infty }{lim}(\parallel z_k-y_k\parallel +\parallel y_k-w_k\parallel +\parallel w_k-x_k\parallel )=0.$$

(40)

The boundedness of $\left\{x_k\right\}$ asserts that there exists a subsequence $\left\{x_{k_j}\right\}$ of $\left\{x_k\right\}$ such that $x_{k_j}\rightharpoonup \widehat{x}$, as $j\to \infty$. Invoking (39), we observe that $w_{k_j}\rightharpoonup \widehat{x}$, as $j\to \infty$. And hence, it follows from Proposition 2 that $\widehat{x}\in VI(C,A)$. Invoking $x^{*}=P_{VI(C,A)}\circ g\left(x^{*}\right)$, we deduce that

$$\underset{k\to \infty }{lim\; sup}\langle g\left(x^{*}\right)-x^{*},x_k-x^{*}\rangle =\underset{j\to \infty }{lim}\langle g\left(x^{*}\right)-x^{*},x_{k_j}-x^{*}\rangle =\langle g\left(x^{*}\right)-x^{*},\widehat{x}-x^{*}\rangle \le 0.$$

(41)

Recalling the definition of $\left\{x_k\right\}$ and the assumption ${lim}_{k\to \infty }{\delta }_k=0$, we infer that

$$\parallel x_{k+1}-x_k\parallel =\parallel (1-{\delta }_k)z_k+{\delta }_{k}g\left(z_k\right)-x_k\parallel \le \parallel z_k-x_k\parallel +{\delta }_k\parallel z_k-g\left(z_k\right)\parallel \to 0,k\to \infty .$$

(42)

Combining (41) with (42), we find that

$$\underset{k\to \infty }{lim\; sup}\langle g\left(x^{*}\right)-x^{*},x_{k+1}-x^{*}\rangle \le 0.$$

(43)

Invoking the conditions ${lim}_{k\to \infty }\frac{{\alpha }_k}{{\delta }_k}\parallel x_k-x_{k-1}\parallel =0$, ${lim}_{k\to \infty }{\delta }_k=0$, ${\sum }_{k=1}^{\infty }{\delta }_k=\infty$, $\kappa \in (0,1)$, we apply Lemma 1 to get that ${lim}_{k\to \infty }\parallel x_k-x^{*}\parallel =0$, that is, ${lim}_{k\to \infty }{x}_k=x^{*}$.

Case 2: Suppose that there is no $k_0\in \mathbb{N}$ such that $\{\parallel x_k-x^{*}{\parallel \}}_{k=k_0}^{\infty }$ is monotonically decreasing. In this case, we can define a mapping $\phi :\mathbb{N}\to \mathbb{N}$ as

$$\phi \left(k\right):=max\{i\in \mathbb{N}:i\le k,\parallel x_i-x^{*}{\parallel }^2\le \parallel x_{i+1}-x^{*}{\parallel }^2\},$$

i.e., $\phi \left(k\right)$ is the largest number i in $\{1,2,\dots ,k\}$ such that $\parallel x_i-x^{*}{\parallel }^2$ increases at $i=\phi \left(k\right)$. Note that $\phi \left(k\right)$ is well-defined for all sufficiently large k. On the other hand, $\phi (·)$ is a nondecreasing sequence such that ${lim}_{k\to \infty }\phi \left(k\right)=\infty$ and the following inequalities hold for all $k\ge 0$,

$$\parallel x_{\phi \left(k\right)+1}{-p\parallel }^2\ge \parallel x_{\phi \left(k\right)}{-p\parallel }^2,\parallel x_{\phi \left(k\right)+1}{-p\parallel }^2\ge {\parallel x_k-p\parallel }^2.$$

(44)

The conditions ${lim}_{k\to \infty }\frac{{\alpha }_k}{{\delta }_k}\parallel x_k-x_{k-1}\parallel =0$ and ${lim}_{k\to \infty }{\delta }_k=0$ entail that ${lim}_{k\to \infty }{\alpha }_k\parallel x_k-x_{k-1}\parallel =0$. Combining (34) and (44) with the boundedness of $\left\{z_k\right\}$, we successfully find that

$$\begin{array}{cc}\hfill & (1-{\lambda }_{\phi \left(k\right)}^2{L}^2){\parallel y_{\phi \left(k\right)}-w_{\phi \left(k\right)}\parallel }^2\hfill \\ \hfill & \le 2{\alpha }_{\phi \left(k\right)}\parallel x_{\phi \left(k\right)}-x_{\phi \left(k\right)-1}\parallel \parallel w_{\phi \left(k\right)}-x^{*}\parallel +{\delta }_{\phi \left(k\right)}(\parallel g\left(x^{*}\right)-x^{*}{\parallel }^2+2\parallel z_{\phi \left(k\right)}-x^{*}\parallel \parallel g\left(x^{*}\right)-x^{*}\parallel )\to 0,\hfill \end{array}$$

as $k\to \infty$. Therefore, it follows from (36) that

$$\underset{k\to \infty }{lim}\parallel y_{\phi \left(k\right)}-w_{\phi \left(k\right)}\parallel =0.$$

(45)

With the help of (45), using the same arguments as in the proof of Case 1, we infer that

$$\underset{k\to \infty }{lim\; sup}\langle g\left(x^{*}\right)-x^{*},x_{\phi \left(k\right)+1}-x^{*}\rangle \le 0.$$

(46)

According to (35) and (44), we have

$$\begin{array}{cc}\hfill \parallel x_{\phi \left(k\right)}-x^{*}{\parallel }^2& \le \frac{2{\alpha }_{\phi \left(k\right)}}{{\delta }_{\phi \left(k\right)}(1-\kappa )}\parallel x_{\phi \left(k\right)}-x_{\phi \left(k\right)-1}\parallel \parallel w_{\phi \left(k\right)}-x^{*}\parallel +\frac{2}{1-\kappa }\langle g\left(x^{*}\right)-x^{*},x_{\phi \left(k\right)+1}-x^{*}\rangle .\hfill \end{array}$$

(47)

Therefore, combining the condition ${lim}_{k\to \infty }\frac{{\alpha }_k}{{\delta }_k}\parallel x_k-x_{k-1}\parallel =0$ with (46) and (47), we have that ${lim\; sup}_{k\to \infty }$$\parallel x_{\phi \left(k\right)}-x^{*}{\parallel }^2\le 0$. And hence, it follows from (44) that $x_k\to x^{*}$, as $k\to \infty$. This completes the proof.  □

3. Numerical Results

In this section, we perform some computational experiments in support of the convergence properties of our proposed methods and compare our methods with Algorithm EAI, see [20].

All programs are written in Matlab version 5.0 and computed on a PC Desktop Intel(R) Core (TM) i5-8250U CPU @1.60GHz. Consider the quadratic programming problem in the form below

$$\left\{\begin{array}{cc}min\hfill & \Lambda =x^T\Theta x+{{\rm Y}}^{T}x\hfill \\ s.t.\hfill & x\in {\mathbb{R}}^n\hfill \\ \hfill & x_i\ge 0,i=1,2,\dots ,n,\hfill \end{array}\right.$$

with the following properties (48)–(50) in a n-dimensional Euclidean space. When $\Theta$ is symmetric and positive-definite in ${\mathbb{R}}^n$, and consequently $A=\nabla \Lambda =\Theta x+{\rm Y}$ is pseudo-monotone and Lipschitz continuous with the constant $L=\parallel \Theta \parallel$. Meanwhile, we choose the parameters ${\lambda }_k=\frac{1}{1.5\parallel \Theta \parallel }$, ${\alpha }_k=\frac{1}{k^2}$, ${\gamma }_k={\delta }_k=\frac{1}{k}$ and ${\beta }_k=\frac{k-1}{2k}$$(k\ge 1)$. We can check that all of conditions in Theorems 1 and 2 are satisfied. We choose randomly initial points $x_0,x_1\in {\mathbb{R}}^n$ in the following experiments. Let us consider the first example [26] with data (48) given by

$$\Theta =\left(\begin{array}{ccccc}1& 2& 2& \dots & 2\\ 2& 5& 6& \dots & 6\\ 2& 6& 9& \dots & 10\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 2& 6& 10& \dots & 4k-3\end{array}\right),{\rm Y}=\left(\begin{array}{c}-1\\ -1\\ -1\\ ⋮\\ -1\end{array}\right).$$

(48)

We apply Algorithm 1 to solve this problem in $H={\mathbb{R}}^5$. We take the iteration number $k=5000$ as the stopping criterion. As depicted in Figure 1, one sees that the optimal solution of this problem is unique ${(1,0,0,0,0)}^T$.

We use the sequence $\left\{E_k\right\}$ defined by $E_k=\parallel x_k-P_C(x_k-{\lambda }_{k}A{x}_k)\parallel (k=1,2,3,\dots )$ to study the convergence of different algorithms in $H={\mathbb{R}}^5$. From the definition of the metric projection, if $\parallel E_k\parallel \le \epsilon$, $x_k$ can be considered as an $\epsilon$-solution of this problem. We take the iteration number $k=100$ as the stopping criterion. To illustrate the computational performance of all the algorithms, the numerical results are shown in Figure 2. From the changing processes of the values of $\left\{E_k\right\}$, we find that Algorithm 2 has a better behavior than Algorithms 1, EAI. It achieves a more stable and higher precision with the number of iterations. Moreover, the convergence of $\left\{E_k\right\}$ to 0, implies that the iterative sequence $\left\{E_k\right\}$ converges to the solution of this test.

Now, we show the second example [27] with the data (49) expressed as

$$\Theta =\left(\begin{array}{ccccccc}4& -1& 0& \dots & 0& 0& 0\\ -1& 4& -1& 0& \dots & 0& 0\\ 0& -1& 4& -1& 0& \dots & 0\\ ⋮& ⋮& \ddots & \ddots & \ddots & \ddots & ⋮\\ 0& \dots & 0& -1& 4& -1& 0\\ 0& 0& \dots & 0& -1& 4& -1\\ 0& 0& 0& \dots & 0& -1& 4\end{array}\right),{\rm Y}=\left(\begin{array}{c}-1\\ -1\\ -1\\ ⋮\\ -1\end{array}\right).$$

(49)

We apply Algorithm 1 to solve this problem in $H={\mathbb{R}}^5$. We set the iteration number $k=5000$ as the stopping criterion. As depicted in Figure 1, one sees that the optimal solution of this problem is unique ${(1,0,0,0,0)}^T$.

This problem is solved for $\Theta$, $50\times 50$ matrix, ${\rm Y}$, 50 vector and $\Theta$, $30\times 30$ matrix, ${\rm Y}$, 30 vector, respectively. We use Algorithms 1 and 2 to solve this problem and we take the iteration number $k=200,100$ as the stopping criterion. The test results are described in Figure 3 and Figure 4. Thus, we obtain the changing processes of $x_1-x_{30}$ with respect to the number of iterations and the running time (x-axis). From this, we find that the iterative sequences generated by Algorithms 1 and 2 converge to a unique solution.

Next, we consider another example [27] with the data (50) written as

$$\Theta =diag(1/k,2/k,\dots ,1),{\rm Y}={(-1,\dots ,-1)}^T.$$

(50)

We take the iteration number $k=1000$ as the stopping criterion in $H={\mathbb{R}}^{25}$. From the results reported in Figure 5, one has shown the changing processes of the values of $x_1$–$x_{25}$ (y-axis) in terms of the number of iterations and the cpu time (x-axis). Accordingly, one sees that Algorithm 1 has a convergent behavior.

4. Conclusions

In this paper, we proposed two inertial extragradient extensions for finding a solution of the pseudo-monotone variational inequalities in the setting of Hilbert spaces. We also established strong convergence theorems of the proposed algorithms. Numerical experiments show that our algorithms enjoy a faster rate of convergence than the one given by Dong et al. [20]. It is worth mentioning that many significant real life problems are generally defined in Banach spaces. Therefore, it is of interest to extend our results to the Banach space, which is more general than the Hilbert space.