Strong convergence of a self - adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and ﬁ xed point problems

: In this paper, we study the problem of ﬁ nding a common solution of the pseudomonotone variational inequality problem and ﬁ xed point problem for demicontractive mappings. We introduce a new inertial iterative scheme that combines Tseng ’ s extragradient method with the viscosity method together with the adaptive step size technique for ﬁ nding a common solution of the investigated problem. We prove a strong convergence result for our proposed algorithm under mild conditions and without prior knowledge of the Lipschitz constant of the pseudomonotone operator in Hilbert spaces. Finally, we present some numerical experiments to show the e ﬃ ciency of our method in comparison with some of the existing methods in the literature.


Introduction
Let H be a real Hilbert space with inner product ⟨⋅ ⋅⟩ , and induced norm ‖⋅‖. In this paper, we consider the variational inequality problem (VIP) of finding a point ∈ p C such that where C is a nonempty closed convex subset of H , and → A H H : is a nonlinear operator. We denote by ( ) VI C A , the solution set of the VIP (1). Variational inequality theory, which was first introduced independently by Fichera [1] and Stampacchia [2], is a vital tool in mathematical analysis, and has a vast application across several fields of study, such as optimisation theory, engineering, physics, operator theory, economics, and many others (see [3][4][5][6] and references therein). Over the years, several iterative methods have been formulated and adopted in solving VIP (1) (see [7][8][9][10][11] and references therein). There are two common approaches to solving the VIP, namely, the regularised methods and the projection methods. These approaches usually require that the nonlinear operator A in VIP (1) has certain monotonicity. In this study, we adopt the projection method and consider the case in which the associated nonlinear operator is pseudomonotone (see definition below)a larger class than monotone mappings. Now, we review some nonlinear operators in nonlinear analysis.  4 5 . However, the converses are not generally true. Moreover, if A is γstrongly monotone and L-Lipschitz continuous, then A is γ L 2inverse strongly monotone (see [12,13]).
The simplest known projection method for solving VIP is the gradient method (GM), which involves a single projection onto the feasible set C per iteration. However, the algorithm only converges weakly under some strict conditions that the operator is either strongly monotone or inverse strongly monotone, but fails to converge if A is monotone. The classical gradient projection algorithm proposed by Sibony [14] is given as follows: where A is strongly monotone and L-Lipschitz continuous, with step size ( ) Korpelevich [15] and Antipin [16] proposed the extragradient method (EGM) for solving VIP (1), thereby relaxing the conditions placed in (7). The initial algorithm proposed by Korpelevich was employed in solving saddle point problems, but was later extended to VIPs in both Euclidean space and infinite dimensional Hilbert spaces. The EGM method is given as follows: Over the years, EGM has been of interest to several researchers. Also, many results and variants have been developed from this method, using the assumptions of Lipschitz continuity, monotonicity, and pseudomonotonicity, see [17][18][19][20] and references therein.
Due to the extensive amount of time required in executing the EGM method, as a result of calculating two projections onto the closed convex set C in each iteration, Censor et al. [8] proposed the subgradient extragradient method (SEGM) in which they replaced the second projection onto C by a projection onto a half-space, thus, making computation easier and convergence rate faster. The SEGM is presented as follows: The authors only obtained a weak convergence result for the proposed method. However, they later introduced a hybrid SEGM in [7] and obtained a strong convergence result. Likewise, Tseng [21], in the bid to improve on the EGM, proposed Tseng's extragradient method (TEGM), which only requires one projection per iteration, as follows: 0 , n C n n n n n n 1 (10) where A is monotone, L-Lipschitz continuous, and The TEGM (10) converges to a weak solution of the VIP with the assumption that ( ) VI C A , is nonempty. The TEGM is also known as the forward-backward method. Recently, some authors have carried out some interesting works on the TEGM (see [22,23] and references therein).
In this work, we consider the inertial algorithm, which is a two-step iteration process and a technique for accelerating the speed of convergence of iterative schemes. The inertial extrapolation technique was derived by Polyak [24] from a dynamic system called the heavy ball with friction. Due to its efficiency, the inertial technique has become a centre of attraction and interest to many researchers in this field. Over the years, researchers have studied the inertial algorithm and applied it to solve different optimisation problems, see [25][26][27][28] and references therein.
Very recently, Tan and Qin [29] proposed the following Tseng's extragradient algorithm for solving pseudomonotone VIP: In this work, we denote the set of fixed points of U by ( ) F U . Our interest in this study is to find a common element of the fixed point set, ( ) F U , and the solution set of the variational inequality, ( ) VI C A , . That is, the problem of finding a point ∈ * x H such that Many algorithms have been proposed over the years and in recent times for solving the common solution problem (13) (see [30][31][32][33][34][35][36][37][38][39][40] and references therein). Common solution problem of this type has drawn the attention of researchers because of its potential application to mathematical models whose constraints can be expressed as FPP and VIP. This arises in areas like signal processing, image recovery, and network resource allocation. An instance of this is in network bandwidth allocation problem for two services in a heterogeneous wireless access networks in which the bandwidth of the services is mathematically related (see [37,41,42] and references therein).
Recently, Cai et al. [22] proposed the following inertial Tseng's extragradient algorithm for approximating the common solution of pseudomonotone VIP and FPP for nonexpansive mappings in real Hilbert spaces: where f is a contraction, T is a nonexpansive mapping, A is pseudomonotone, L-Lipschitz and sequentially weakly continuous, and ( ) ∈ ψ 0, L 1 . They proved a strong convergence result for the proposed algorithm under some suitable conditions. One of the major drawbacks of Algorithm (14) is the fact that the step size ψ of the algorithm depends on the Lipschitz constant of the cost operator. In many cases, this Lipschitz constant is unknown or even difficult to estimate. This makes it difficult to implement algorithms of this nature.
Very recently, Thong and Hieu [23] proposed an iterative scheme for finding a common element of the solution set of monotone variational inequality and set of fixed points of demicontractive mappings as follows: where A is monotone and L-Lipschitz continuous, U is a demicontractive mapping such that − I U is demiclosed at zero, and f is a contraction. The authors proved a strong convergence result under suitable conditions for the proposed method.
Motivated by the above results and the ongoing research activities in this direction, in this paper our aim is to introduce an effective iterative technique, which employs the efficient combination of the inertial technique, TEGM together with the viscosity method for finding a common solution of FPP of demicontractive mappings and pseudomonotone VIP with Lipschitz continuous and sequentially weakly continuous operator in Hilbert spaces. In line with this goal, we construct an algorithm with the following features: (i) Our algorithm approximates the solution of a more general class of VIP and FPP.
(ii) The proposed method only requires one projection per iteration onto the feasible set, which guarantees the minimal cost of computation.
(iii) Moreover, our method is computationally efficient. It employs an efficient self-adaptive step size technique which makes the algorithm independent of the Lipschitz constant of the cost operator. (iv) We employ the combination of the inertial technique together with the viscosity method, which are two of the efficient techniques for accelerating the rate of convergence of iterative schemes. (v) We prove a strong convergence theorem for the proposed algorithm without following the conventional "two-cases" approach often employed by researchers (e.g. see [22,23,29,[43][44][45]). This makes our results in this paper to be more concise and precise.
Furthermore, by several numerical experiments, we demonstrate the efficiency of our proposed method over many other existing methods in related literature.
The remainder of this paper is organised as follows. In Section 2, useful definitions and lemmas employed in the study are presented. In Section 3, we present the proposed algorithm and highlight some of its notable features. Section 4 presents the convergence analysis of the proposed method. In Section 5, we carry out some numerical experiments to illustrate the computational advantage of our method over some of the existing methods in the literature. Finally, in Section 6 we give a concluding remark.

Preliminaries
Let H be a real Hilbert space and C be a nonempty closed convex subset of H . We denote the weak and strong convergence of sequence The metric projection [46,47], . Also, the mapping P C is firmly nonexpansive, i.e.
. Some results on the metric projection map are given below.
or equivalently Remark 2.4. It is known that every strictly pseudocontractive mapping with a nonempty fixed point set is demicontractive. The class of demicontractive mappings includes all the other classes of mappings defined above (see [23]).
Next, we give some examples of the class of demicontractive mappings, as shown in [23,50].
Then T is demicontractive.
Then T is a demicontractive map that is neither quasi-nonexpansive nor strictly pseudocontractive.
We have the following lemmas which will be employed in our convergence analysis.
Lemma 2.6. [25] For each ∈ x y H , , and ∈ δ , we have the following results: , and { } b n be a sequence of real numbers. Assume that is a nonlinear operator with ( ) ≠ F T 0. Then, − I T is said to be demiclosed at zero if for any { } x n in H, the following implication holds: Lemma 2.9. [53] Assume that D is a strongly positive bounded linear operator on a Hilbert space H with coefficient > γ 0 and

Proposed algorithm
In this section, we propose an inertial viscosity-type Tseng's extragradient algorithm with self adaptive step size and highlight some of its important features. We establish the convergence of the algorithm under the following conditions: Condition A (A1) The feasible set C is closed, convex, and nonempty.
The mapping A is pseudomonotone, L-Lipschitz continuous on H , and sequentially weakly continuous on C.
is a strongly positive bounded linear operator with coefficient γ.
Now, the algorithm is presented as follows: Algorithm 3.1. Inertial TEGM with self-adaptive stepsize _____________________________________________________________________________________________ Step 0. Given , and set = n 1.
Step 1. Given the ( − n 1)th and nth iterates, choose δ n such that ≤ ≤ ∀ ∈ δ δ n 0ˆ, n n with δ n defined by Step 2. Compute n n n n n 1 Step 3. Compute Step 5. Compute n n n n n n n n 1 Step 6. Compute

Convergence analysis
First, we establish some lemmas which will be employed in the convergence analysis of our proposed algorithm.       (27) So, from (25), (26), and (27), we have Now, from ( ) ∈ p VI C A , , we have that Combining (28) and (29), we have that Proof. We divide the proof of Theorem 4.4 as follows: Claim 1. The sequence { } x n generated by Algorithm 3.1 is bounded. First, we show that (   By the condition on β n , from this we obtain Also, by definition of r n and triangle inequality, 1 .      Using the Cauchy-Schwartz inequality and Lemma 2.6, we obtain where   Consequently, we obtain Again, from Claim 2 we obtain Applying (38) and the fact that By the conditions on the control parameters, we obtain Following similar argument, from Claim 2 we have From (24) and (39), we obtain Combining (39) and (41) is a τ-demicontractive map. Let { } x n be a sequence generated as follows: Algorithm 4.6. _____________________________________________________________________________________________ Step 0. Given Step 2. Compute n n n n n 1 (52) Step 3. Compute the projection: Step 6. Compute  [23], our result in Corollary 4.5 employs inertial technique to speed up the rate of convergence of the algorithm. (vi) As shown in our convergence analysis, we did not adopt the conventional "two cases" approach employed in several papers to prove strong convergence. Our procedure is more concise and easy to comprehend.

Numerical examples
In this section, we proceed to perform two numerical experiments to show the computational efficiency of our  . We plot the graphs of errors against the number of iterations in each case. The numerical results are reported in Figure 1 and Table 1.

Conclusion
We studied the pseudomonotone VIP with a fixed point constraint. We introduced a new inertial TEGM with an adaptive step size for approximating a solution of the pseudomonotone VIP, which is also a fixed point of demicontractive mappings. We proved strong convergence results for the proposed algorithm without the knowledge of the Lipschitz constant of the cost operator. Finally, we presented several numerical experiments to demonstrate the efficiency of our proposed method in comparison with some of the existing methods in the literature.