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

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.

The variational inequality problem (VIP) is the problem to find a point \(x^{*}\in C\) such that The solution set of VIP is denoted by \(VI(C,A)\). The variational inequality problem is an important branch of the nonlinear problem and it has received a lot of attentions by many authors in recent years (see [1‐3] and the references therein). Under appropriate conditions, there are two general approaches for solving the variational inequality problem, one is the regularized method and the other is the projection method. Now, we mainly study the projection method.

For every point \(x\in H\), there exists a unique point in C such that where \(P_{C}:H\rightarrow C\) is called the metric projection from H into C.

We know that the variational inequality problem can be turned into the fixed point problem, which means (1) is equivalent to where \(P_{C}:H\rightarrow C\) is the metric projection and \(\lambda >0\). Thus we generate \(\{x_{n}\}\) in the following manner: This simple algorithm is an extension of the projection gradient method. However, the convergence of this method requires a slight assumption that the operator \(A:H\rightarrow H\) is strongly monotone or inverse strongly monotone.

To avoid this strong assumption, Korpelevich [4] proposed an algorithm which was called the extragradient method: for each \(n=1,2,\ldots \) , where \(\lambda \in (0,1/L)\). At the beginning, this method was used to solve saddle point problems. Soon, this method was extended to Euclidean spaces and Hilbert spaces. In particular, this method only requires that the operator 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.

To our knowledge, there were four kinds of methods to overcome this drawback. The first one was the modification of the extragradient method by Tseng [5] who proposed it in 2000 with the following remarkable scheme: where 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: where 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.

Before explaining the third method, let us take a look at the inertial method. In 2001, Alvarez and Attouch [7] applied the inertial technique to obtain an inertial proximal method to solve the problem of finding zero of a maximal monotone operator, which works as follows: where \(x_{n-1}, x_{n}\in H\), \(\theta _{n}\in [0,1)\) and \(\lambda _{n}>0\). It also can be written in the following form: where \(J^{A}_{\lambda _{n}}\) is the resolvent of 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: for each \(k\geq 1\), where \(\gamma \in (0,2)\), \(\lambda >0\), This algorithm incorporate the inertial terms in the projection and contraction algorithm, which does not need the summability condition for the sequence. The fourth one was a self-adaptive algorithm which was based on Tseng’s extragradient method [9] and it was proposed by Duong Viet Thong and Dang Van Hieu [9] in 2017. The algorithm is described as follows.

It is worth mentioning that the Algorithm 1 does not require one to know the Lipschitz constant of the operator 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.

In 2016, Mainge and Gobinddass [10] got \(x_{n+1}\) by the following algorithm: where \(\theta _{n}=\frac{\lambda _{n}}{\delta \lambda _{n-1}}\), \(\lambda _{n}k_{n}\leq \varepsilon \delta (\sqrt{2}-1)\), \(\lambda _{n}\leq k\lambda _{n-1}(\delta +\frac{\lambda _{n-1}}{\lambda _{n-2}})^{ \frac{1}{2}}\), \(\{\lambda _{n}\}\subset [\overline{\mu },\overline{ \nu }]\) and In this iterative algorithm, it does not require either additional projection for the determination of the step-sizes or the knowledge of the Lipschitz constant of the operator.

The fixed point problem is the problem to find \(x^{*}\in H\) such that where \(x^{*}\) is called a fixed point of \(T:H\rightarrow H\). The set of fixed points of T is denoted by \(\operatorname{Fix}(T)\). Recently, many iterative methods have been proposed (see [6, 11‐21] 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.

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 \(\omega _{w}(x _{n})\).

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

Next, we present some concepts of an operator.

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.

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.

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

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].