Strong convergence inertial projection algorithm with self-adaptive step size rule for pseudomonotone variational inequalities in Hilbert spaces


 In this paper, we introduce a new algorithm for solving pseudomonotone variational inequalities with a Lipschitz-type condition in a real Hilbert space. The algorithm is constructed around two algorithms: the subgradient extragradient algorithm and the inertial algorithm. The proposed algorithm uses a new step size rule based on local operator information rather than its Lipschitz constant or any other line search scheme and functions without any knowledge of the Lipschitz constant of an operator. The strong convergence of the algorithm is provided. To determine the computational performance of our algorithm, some numerical results are presented.


Introduction
This paper studies the problem of classic variational inequalities [1,2]. The variational inequality problem (VIP) for a mapping → : , which is formulated in the following way: where is a nonempty, convex and closed subset of a real Hilbert space and ⟨ ⟩ . , . and ∥ ∥ . represent an inner product and the induced norm in , respectively. Moreover, , are the sets of real numbers and natural numbers, respectively. It is important to note that the problem (VIP) is equivalent to solve the following problem: The concept of variational inequalities has been used as an important tool for covering a large number of topics, i.e., physics, engineering, economics and optimization theory. This was introduced by Stampacchia [1] in 1964. This is a significant mathematical design that unifies several key topics of applied mathematics, such as the network equilibrium problem, the necessary optimality conditions, the complementarity problems and the systems of nonlinear equations (for more details [3][4][5][6][7][8][9][10][11][12]). On the other hand, the projection algorithms are important to find the numerical solution of variational inequalities. Many experts have introduced and considered many projection algorithms to study the variational inequality problems (see for more details [13][14][15][16][17][18][19][20][21][22][23][24][25]) and others in [26][27][28]. Korpelevich [13] and Antipin [29] introduced the following extragradient algorithm: Recently, the subgradient extragradient algorithm was provided by Censor et al. [15] for solving the problem (VIP) in a real Hilbert space. Their algorithm takes the form It is important to note that the above well-established algorithm carries two serious drawbacks, the first is the fixed constant step size that requires the knowledge or approximation of the Lipschitz constant of the relevant operator and it only converges weakly in Hilbert spaces. From the computational point of view, it might be problematic to use fixed step size, and hence the convergence rate and usefulness of the algorithm could be affected.
Yang et al. [30] proposed two explicit subgradient extragradient methods to solve monotone variational inequalities. An iterative sequence { } u n was generated in the following way: Algorithm A. (ii) Compute iterative sequence { } u n for ≥ n 1 as follows: n n n n n n n n n n n n 1 0 n n n n n n (iii) Update the step size rule in the following way: otherwise. The main objective of this paper is to set up an inertial-type algorithm that is used to improve the convergence rate of the iterative sequence. Such algorithms have been previously established due to the oscillator equation with a damping and conservative force restoration. This second-order dynamical system is called a heavy friction ball, which was originally studied by Polyak in [31]. The main feature of inertialtype algorithms is that they can use the two previous iterations to obtain the next iteration. Numerical results confirm that inertial terms normally improve the performance of the algorithm in terms of the number of iterations and elapsed time in this context.
So there is an important question: "Is it possible to establish a new inertial-like strongly convergent extragradient-type algorithm with a monotone variable step size rule?" In this research, we provide a positive answer to the above question, i.e., the gradient algorithm indeed establishes a strong convergence sequence by maintaining variable step size rule for solving problem (VIP) combined with pseudomonotone mappings. Motivated by the works of Censor et al. [15] and Polyak [31], we introduce a new inertial extragradient-type algorithm to figure out the problem (VIP) in the situation of an infinite-dimensional real Hilbert space.
Specifically, our key contributions to this paper are as follows: ⊙ We introduce an inertial subgradient extragradient algorithm with the use of a variable monotone step size rule independent of the Lipschitz constant to figure out pseudo-monotone VIPs. ⊙ We also provide numerical experiments corresponding to the proposed algorithms for the verification of theoretical findings and compare them with the results in Algorithm 1 in [30] and Algorithm 2 in [30].
Our numerical data have shown that the proposed algorithms are useful and performed better compared to the existing ones in most situations.
The rest of the paper is arranged as follows: Section 2 consists of the necessary definitions and fundamental lemmas needed in the article. Section 3 consists of an inertial-type iterative scheme and convergence analysis theorem. Section 4 provides numerical results to explain the performance of the new algorithm and to compare them with other existing algorithms.

Preliminaries
In this section of the text, we have written a number of significant identities and related lemmas and definitions.
Next, we list some of the important properties of the projection mapping. is a metric projection. Then, we have and ℓ ∈ , the following inequalities hold: is a pseudomonotone and continuous mapping. Then, * q is a solution of the problem (VIP) if and only if * q is a solution of the following problem:

Inertial two-step proximal-like algorithm and convergence analysis
In this section, we introduce an inertial-type subgradient extragradient algorithm which incorporates the new step size rule and the inertial term as well as provides both strong convergence theorems.
The following main result is outlined as follows:
This implies that the sequence { } ζ n has a lower bound □ In order to study the strong convergence, the following conditions are satisfied: (A1) The solution set of problem (VIP) denoted by Ω is nonempty; (A2) An operator → : is said to be pseudomonotone, i.e., , ; is said to be sequentially weakly continuous, i.e., { ( )} u n converges weakly to ( ) u for every sequence { } u n converges weakly to u.
Proof. It is given that Thus, we have which implies that n n n n n n n n n n n n n n n 2 n n (9) Combining expressions (7) and (9), we obtain Since * q is the solution of problem (VIP), we have Due to the pseudomonotonicity of on , we get By substituting = ∈ v v n , we obtain n n Thus, we have Strong convergence inertial projection method  115 Combining expressions (10) and (11), we get Combining expressions (12) and (13), we obtain □ Theorem 3.3. Let { } u n be a sequence generated by Algorithm 1 and satisfy the conditions ( 1)-( 4). Then, Thus, there exists a finite number ∈ n 1 such that This implies that It is given in expression (16) that By the use of definition of { } η n and inequality (17), we obtain n n n n n n n n n n n n n n n n n n where Combining expressions (16) and (19), we obtain Thus, we conclude that { } u n is a bounded sequence. Indeed, by expression (19) we have where for some > M 0.
2 By using the expressions (14) with (21), we have The rest of the proof will be divided into the following two parts: Next, we have compute The following provides that The above expression guarantees that the sequences { } η n and { } v n are also bounded. By the use of reflexivity of and the boundedness of { } u n guarantees that there exists a subsequence { } u nk such that { } ⇀ ∈ u û nk as → +∞ k . Next, we have to prove that ∈ û Ω. It is given that The inequality described above implies that Furthermore, we obtain n n n n n n n n n The sequence { ( )} η nk is bounded due to the boundedness of sequence { } η nk . By the use of Furthermore, we have Combining expressions (33) and (34), we obtain Consider a sequence of positive numbers { } ε k that is decreasing and converges to zero. For each k, we denote m k by the smallest positive integer such that It is obvious that { } m k is an increasing sequence because { } ε k is a decreasing sequence.
Case A: Let there exists a subsequence η nm k j Hence ∈ û , therefore we obtain ∈ û Ω.
By the use of Minty Lemma 2.5, we infer ∈ û Ω. It is given that = ( ) * q P 0 Ω and by using Lemma 2.1(ii), we have Next, we have to Taking into account expression (18), we have Due to → β 0 mk , we can conclude the following: The above implies that Next, we have to compute This implies that   Table 1).      It is easy to verify that Q is symmetric and positive definite on 4 and consequently f is pseudo-convex on . Hence, ∇f is pseudo-monotone. Using the quotient rule, we obtain .       Strong convergence inertial projection method  125