# Self-adaptive subgradient extragradient method with inertial modification for solving monotone variational inequality problems and quasi-nonexpansive fixed point problems

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## Abstract

In this paper, we introduce a new algorithm with self-adaptive method for finding a solution of the variational inequality problem involving monotone operator and the fixed point problem of a quasi-nonexpansive mapping with a demiclosedness property in a real Hilbert space. The algorithm is based on the subgradient extragradient method and inertial method. At the same time, it can be considered as an improvement of the inertial extragradient method over each computational step which was previously known. The weak convergence of the algorithm is studied under standard assumptions. It is worth emphasizing that the algorithm that we propose does not require one to know the Lipschitz constant of the operator. Finally, we provide some numerical experiments to verify the effectiveness and advantage of the proposed algorithm.

## Keywords

Variational inequality problem Fixed point problem Extragradient method Subgradient extragradient method Inertial method Self-adaptive method## 1 Introduction

Throughout this paper, let *H* be a real Hilbert space with the inner product \(\langle \cdot ,\cdot \rangle \) and norm \(\|\cdot \|\). Let *C* be a nonempty, closed and convex subset of *H*. Let \(\mathbb{N}\) and \(\mathbb{R}\) be the sets of positive integers and real numbers, respectively.

*C*such that

*H*into

*C*.

*A*is monotone and

*L*-Lipschitz continuous in a Hilbert space. If \(VI(C,A)\neq \emptyset \), the sequence \(\{x_{n}\}\) generated by (4) converges weakly to an element of \(VI(C,A)\).

However, the extragradient method needs to calculate two projections from *H* onto the closed convex set *C* and it is applicable to the case that \(P_{C}\) has a closed form which means that \(P_{C}\) has an explicit expression. In fact, in some cases, the projection onto the nonempty closed convex subset *C* might be difficult to calculate. To overcome this drawback, it has received great attentions by many authors who had improved it in various ways.

*A*is monotone,

*L*-Lipschitz continuous and \(\lambda \in (0,1/L)\). From (5), we find using this method one only needs to calculate one projection, which is simpler than (4). The second one was the subgradient extragradient method which was proposed by Censor et al. [6] in 2011:

*A*is monotone,

*L*-Lipschitz continuous and \(\lambda \in (0,1/L)\). The key operation of the subgradient extragradient method replaces the second projection onto

*C*of the extragradient method by a projection onto a special constructible half-space, which significantly reduces the difficulty of calculations.

*A*with parameter \(\lambda _{n}\) and the inertia is induced by the term \(\theta _{n}(x _{n}-x_{n-1})\). Recently, considerable interest has been shown in studying the inertial method by many authors. They constructed fast iterative algorithms by using inertial method. The third method which was studied by Q.L. Dong et al. [8] in 2017:

*A*, which is different from the other three algorithms. If \(VI(C,A)\neq \emptyset \), the sequences \(\{x_{n}\}\) generated by (5), (6), (8) and Algorithm 1 all converge weakly to an element of \(VI(C,A)\). For Algorithm 1, it does not require to know the Lipschitz constant, but the step size may involve computation of additional projections.

*T*is denoted by \(\operatorname{Fix}(T)\). Recently, many iterative methods have been proposed (see [6, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] and the references therein) for finding a common element of \(\operatorname{Fix}(T)\) and \(VI(C,A)\) in a real Hilbert space.

In this paper, motivated and inspired by the above results, we introduce a new algorithm with self-adaptive subgradient extragradient method and inertial modification for finding a solution of the variational inequality problem involving monotone operator and the fixed point problem of a quasi-nonexpansive mapping with a demiclosedness property in a real Hilbert space. Then the weak convergence theorem will be proved in Sect. 3.

This paper is organized as follows. In Sect. 2, we list some lemmas which will be used for further proof. In Sect. 3, we proposed a new algorithm, then the weak convergence theorem is analyzed. In Sect. 4, we give some numerical examples to illustrate the efficiency and advantage of our algorithm.

## 2 Preliminaries

In this section, we introduce some lemmas which will be used in this paper. Assume *H* is a real Hilbert space and *C* is a nonempty closed convex subset of *H*. In the following of the paper, we use the symbol \(x_{n}\rightarrow x\) to denote the strong convergence of the sequence \(\{x_{n}\}\) to *x* as \(n\rightarrow \infty \) and use the symbol \(x_{n}\rightharpoonup x\) to denote the weak convergence of the sequence \(\{x_{n}\}\) to *x* as \(n\rightarrow \infty \). If there exists a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) converging weakly to a point *z*, then *z* is called a weak cluster point of \(\{x_{n}\}\) and the set of all weak cluster points of \(\{x_{n}\}\) is denoted by \(\omega _{w}(x _{n})\).

### Lemma 2.1

([22])

*Let*

*H*

*be a real Hilbert space*,

*for each*\(x,y\in H\)

*and*\(\lambda \in \mathbb{R}\),

*we have*

- (i)
\(\|x+y\|^{2}=\|x\|^{2}+\|y\|^{2}+2\langle x,y\rangle \);

- (ii)
\(\|\lambda x+(1-\lambda )y\|^{2}=\lambda \|x\|^{2}+(1-\lambda )\|y\|^{2}-\lambda (1-\lambda )\| x-y\|^{2}\).

In the following, we gather some characteristic properties of \(P_{C}\).

### Lemma 2.2

([23])

*Let*

*H*

*be a real Hilbert space and*

*C*

*be a nonempty closed subset of*

*H*.

*Then*

- (i)
\(\|P_{C}x-P_{C}y\|^{2}\leq \langle x-y,P_{C}x-P_{C}y\rangle \), \(\forall x,y\in H\);

- (ii)
\(\|x-P_{C}x\|^{2}+\|y-P_{C}x\|^{2}\leq \| x-y\|^{2}\), \(\forall x\in H\), \(y\in C\).

### Lemma 2.3

*Let**H**be a real Hilbert space and**C**be a nonempty closed subset of**H*. *Given*\(x\in H\)*and*\(z\in C\), *then*\(z=P_{C}x\)*if and only if there hold the inequality*\(\langle x-z,y-z \rangle \leq 0\), \(\forall y\in C\).

Next, we present some concepts of an operator.

### Definition 2.4

([24])

- (i)monotone, if$$ \langle x-y,Ax-Ay\rangle \geq 0,\quad \forall x,y\in H; $$
- (ii)
*L*-Lipschitz continuous with \(L>0\), if$$ \Vert Ax-Ay \Vert \leq L \Vert x-y \Vert ,\quad \forall x,y\in H; $$ - (iii)nonexpansive, if$$ \Vert Ax-Ay \Vert \leq \Vert x-y \Vert ,\quad \forall x,y\in H; $$
- (iv)quasi-nonexpansive, ifwhere \(Fix(A)\neq \emptyset \).$$ \Vert Ax-p \Vert \leq \Vert x-p \Vert ,\quad \forall x\in H, p\in Fix(A), $$

### Remark 2.5

([25])

It is well that every nonexpansive mapping with a nonempty set of fixed point is quasi-nonexpansive. However, a quasi-nonexpansive mapping may not be a nonexpansive mapping.

### Lemma 2.6

([23])

*Assume that*\(T:H\rightarrow H\)

*is a nonlinear operator with*\(\operatorname{Fix}(T)\neq \emptyset \).

*Then*\(I-T\)

*is said to be demiclosed at zero if for any*\(\{x_{n}\}\)

*in*

*H*,

*the following implication holds*:

### Remark 2.7

We know that the Lemma 2.6 is clearly established when the operator *T* is nonexpansive. However, there exists a quasi-nonexpansive mapping *T* but \(I-T\) is not demiclosed at zero. Therefore, in this paper, we need to emphasize that \(T:H\rightarrow H\) is a quasi-nonexpansive mapping such that \(I-T\) is demiclosed at zero.

### Example 1

*H*be the line real and \(C=[0,\frac{3}{2}]\). Define the operator

*T*on

*C*by

*T*is quasi-nonexpansive.

### Lemma 2.8

([7])

*Let*\(\{\varphi _{n}\}\), \(\{\delta _{n}\}\)

*and*\(\{\alpha _{n}\}\)

*be sequences in*\([0,+\infty )\)

*such that*

*and there exists a real number*

*α*

*with*\(0\leq \alpha _{n}\leq \alpha <1\)

*for all*\(n\in \mathbb{N}\).

*Then the following hold*:

- (i)
\(\sum^{+\infty }_{n=1}[\varphi _{n}-\varphi _{n-1}]_{+}<+\infty \),

*where*\([t]_{+}:=\max \{t,0\}\); - (ii)
*there exists*\(\varphi ^{*}\in [0,+\infty )\)*such that*\(\lim_{n\rightarrow +\infty }\varphi _{n}=\varphi ^{*}\).

### Lemma 2.9

([26])

*Let*\(A:H\rightarrow H\)*be a monotone and**L*-*Lipschitz continuous mapping on**C*. *Let*\(S=P_{C}(I- \mu A)\), *where*\(\mu >0\). *If*\(\{x_{n}\}\)*is a sequence in**H**satisfying*\(x_{n}\rightharpoonup q\)*and*\(x_{n}-Sx_{n}\rightarrow 0\), *then*\(q\in VI(C,A)=\operatorname{Fix}(S)\).

### Lemma 2.10

([27])

*Let*

*C*

*be a nonempty closed and convex subset of a real Hilbert space*

*H*

*and*\(\{x_{n}\}\)

*be a sequence in*

*H*.

*The following two properties hold*:

- (i)
\(\lim_{n\rightarrow \infty }\|x_{n}-x\|\)

*exists for each*\(x\in C\); - (ii)
\(\omega _{w}(x_{n})\subset C\).

*Then the sequence*\(\{x_{n}\}\)

*converges weakly to a point in*

*C*.

## 3 Main results

In this section, we propose a new iterative algorithm with self-adaptive method for solving monotone variational inequality problems and quasi-nonexpansive fixed point problems in a Hilbert space. Meanwhile, we combine subgradient extragradient method and inertial modification for the algorithm. Under the assumption \(\operatorname{Fix}(T)\cap VI(C,A)\neq \emptyset \), we prove the weak convergence theorem. Let *H* be a real Hilbert space. Let *C* be a nonempty closed convex subset in *H*. Let \(A:H\rightarrow H\) be a monotone and *L*-Lipschitz continuous operator. In particular, the information of the Lipschitz constant *L* does not require to be known. Let \(T:H\rightarrow H\) be a quasi-nonexpansive mapping such that \(I-T\) is demiclosed at zero. The algorithm is described as follows.

Before giving the theorem and its proof, we propose several useful lemmas firstly.

### Lemma 3.1

*The sequence*\(\{\lambda _{n}\}\)

*generated by Algorithm*2

*is a monotonically decreasing sequence*,

*and its lower bound is*\(\min \{\frac{\mu }{L},\lambda _{0}\}\).

### Proof

It is obvious that the sequence \(\{\lambda _{n}\}\) is a monotonically decreasing sequence.

*A*is

*L*-Lipschitz continuous with \(L>0\), we have

Clearly, the lower bound of the sequence \(\{\lambda _{n}\}\) is \(\min \{\frac{\mu }{L},\lambda _{0}\}\). □

### Lemma 3.2

*If*\(w_{n}=y_{n}=x_{n+1}\), *then*\(w_{n} \in \operatorname{Fix}(T)\cap VI(C,A)\).

### Proof

If \(w_{n}=y_{n}\), we have \(w_{n}\in VI(C,A)\).

Besides, since \(w_{n}=y_{n}\), \(y_{n}=P_{C}(w_{n}-\lambda _{n} Aw_{n})\), according to Lemma 2.3, we have \(\langle w_{n}-\lambda _{n} Aw_{n}-y _{n},x-y_{n}\rangle \leq 0\), \(\forall x\in C\). Since \(w_{n}=y_{n}\), \(z_{n}=P_{T_{n}}(w_{n}-\lambda _{n}Ay_{n})\), where \(T_{n}=\{x\in H| \langle w_{n}-\lambda _{n} Aw_{n}-y_{n},x-y_{n}\rangle \leq 0\}\), we have \(y_{n}=z_{n}\).

Therefore, \(w_{n}\in \operatorname{Fix}(T)\cap VI(C,A)\). □

### Lemma 3.3

*Let*\(\{z_{n}\}\)

*be a sequence generated by Algorithm*2,

*then*,

*for all*\(p\in VI(C,A)\),

*and for*

*n*

*sufficiently large*,

*we have*

### Proof

*A*is monotone and \(\lambda _{n}>0\), we have

### Theorem 3.4

*Assume that the sequence*\(\{\alpha _{n}\}\)*is non*-*decreasing such that*\(0\leq \alpha _{n}\leq \alpha \leq \frac{1}{4}\)*and the sequence*\(\{\beta _{n}\}\)*is a sequence of real numbers such that*\(0<\beta \leq \beta _{n}\leq \frac{1}{2}\). *Then the sequence*\(\{x_{n}\}\)*generated by Algorithm *2 *converges weakly to an element of*\(\operatorname{Fix}(T)\cap VI(C,A)\).

### Proof

Let \(p\in \operatorname{Fix}(T)\cap VI(C,A)\).

From Lemma 3.3, we have \(\exists N\geq 0\), \(\forall n>N\), \(\|z_{n}-p\| \leq \|w_{n}-p\|\).

*T*is quasi-nonexpansive, by Lemma 2.1, we have \(\forall n>N\)

Put \(\varGamma _{n}:=\|x_{n}-p\|^{2}-\alpha _{n}\|x_{n-1}-p\|^{2}+2\alpha _{n}\|x_{n}-x_{n-1}\|^{2}\).

We have \(0\leq \alpha _{n}\leq \alpha \leq \frac{1}{4}\), \(-(2\alpha _{n+1}-1+\alpha _{n})\geq \frac{1}{4}\).

So \(\varGamma _{n+1}-\varGamma _{n}\leq -\delta \|x_{n+1}-x_{n}\|^{2}\leq 0\), where \(\delta =\frac{1}{4}\), which implies that the sequence \(\{\varGamma _{n}\}\) is non-increasing.

This implies that the sequence \(\{x_{n}\}\) is bounded.

Since \(\{x_{n}\}\) is bounded, there exist a subsequence \(\{x_{n_{k}} \}\) of \(\{x_{n}\}\) and \(q\in H\) such that \(x_{n_{k}}\rightharpoonup q\).

So, by (32) we have \(\omega _{n_{k}}\rightharpoonup q\) and by (37) we have \(z_{n_{k}}\rightharpoonup q\).

Since \(z_{n_{k}}\rightharpoonup q\) and \(I-T\) is demiclosed at zero, by Lemma 2.6, we have \(q\in \operatorname{Fix}(T)\).

By Lemma 2.9, we have \(q\in VI(C,A)\).

Therefore, \(q\in \operatorname{Fix}(T)\cap VI(C,A)\).

By Lemma 2.10, we get the conclusion that the sequence \(\{x_{n}\}\) converges weakly to an element of \(\operatorname{Fix}(T)\cap VI(C,A)\).

This completes the proof. □

## 4 Numerical experiments

In this section, we give some numerical examples to illustrate the efficiency and advantage of our algorithm in comparisons with the well-known algorithm. We compare Algorithm 2 with the weakly convergent Algorithm 1 [19].

We choose \(\alpha _{n}=\frac{1}{4}\), \(\beta _{n}=\frac{1}{2}\), \(\mu =\frac{1}{2}\), \(\lambda _{0}=\frac{1}{7}\). The starting point is \(x_{0}=x_{1}=(1,1,\ldots ,1)\in \mathfrak{R}^{m}\). In order to show the converges of the algorithm, we illustrate the behavior of the sequence \(D_{n}=\|x_{n}-x^{*}\|^{2}\), \(n=0,1,2,\ldots \) , when the execution time in second elapses where \(x^{*}\) is the solution of the problem and \(\{x_{n}\}\) is the sequence generated by the algorithms. Now we introduce the examples in detail.

### Example 2

*A*be a Lipschitz continuous and monotone mapping. Let

*T*be a quasi-nonexpansive mapping. Assume \(\operatorname{Fix}(T) \cap VI(C,A)\neq \emptyset \) and \(C=[-2,5]\), \(H=\mathbb{R}\). Let

*A*and

*T*be given by

In the following, let us verify if *A* and *T* meet the requirements of the topic.

Therefore, \(\|Ax-Ay\|\leq L\|x-y\|\), where \(L=2\) and \(\langle Ax-Ay,x-y \rangle \geq 0\). Therefore, *A* is *L*-Lipschitz continuous and monotone.

Second, for \(Tx=\frac{x}{2}\sin x\), if \(x\neq 0\) and \(Tx=x\), then we have \(x=\frac{x}{2}\sin x\), and \(\sin x=2\), which is impossible. Therefore, we obtain \(x=0\), which means \(\operatorname{Fix}(T)=\{0\}\).

*T*is quasi-nonexpansive.

*T*is not a nonexpansive mapping.

*A*and

*T*meet the requirements of the topic. The numerical results for the example are shown in Fig. 1.

From Fig. 1, we can see that the Algorithm 2 converges for a shorter time than the previously studied Algorithm 1 [19].

### Example 3

*N*is a \(m\times m\) matrix,

*S*is a \(m\times m\) skew-symmetric matrix,

*D*is a \(m\times m\) diagonal matrix which its diagonal entries are nonnegative, and \(q\in \mathfrak{R}^{m}\) is a vector, therefore

*M*is positive definite. The feasible set is

It is obvious that *A* is monotone and Lipschitz continuous. For experiments, *q* is equal to zero vector, all the entries of *N*, *S* are generated randomly and uniformly in \([-2,2]\), and the diagonal entries of *D* are in \((0,2)\).

According to Figs. 2, 3, and 4, we have confirmed that the proposed algorithm have the competitive advantages over the existing Algorithm 1 [19].

## 5 Conclusion

In this paper, we introduce a new algorithm with self-adaptive method for finding a solution of the variational inequality problem involving monotone operator and the fixed point problem of a quasi-nonexpansive mapping with a demiclosedness property in a real Hilbert space. We combine a subgradient extragradient method and inertial modification for the algorithm. Under some suitable conditions, we have proved the weak convergence of the algorithm. In particular, it is worth emphasizing that the algorithm that we propose does not need any additional projections of the Lipschitz constant. Finally, some numerical experiments are performed to verify the convergence of the algorithm and compared with previously known Algorithm 1 [19].

## Notes

### Authors’ contributions

All the authors read and approved the final manuscript.

### Funding

This work was supported by the Financial Funds for the Central Universities (No. 3122018L004) and Scientific research project of Tianjin Municipal Education Commission (No. 2018KJ253).

### Competing interests

The authors declare that they have no competing interests.

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