Solving Bilevel Quasimonotone Variational Inequality Problem In Hilbert Spaces

: In this paper, we propose and study a Bilevel quasimonotone Variational Inequality Problem (BVIP) in the framework of Hilbert space. We introduce a new modiﬁed inertial iterative technique with self-adaptive step size for approximating a solution of the BVIP. In addition, we established a strong convergence result of the proposed iterative technique with adaptive step-size conditions without prior knowledge of Lips-chitz’s constant of the cost operators as well as the strongly monotonicity coeﬃcient under some standard mild assumptions. Finally, we provide some numerical experiments to demonstrate the eﬃciency of our proposed methods in comparison with some recently announced results in the literature.


Introduction
Let H be a real Hilbert space with the inner product •, • and the induced norm • , C a nonempty closed convex subset of H and F : H → H be a nonlinear operator.The classical Variational Inequality Problem (VIP) is formulated as: (1.1) The notion of VIP was introduced independently by Stampacchia [36] and Fichera [13,14] for modeling problems arising from mechanics and for solving Signorini problem.It is well-known that many problems in economics, mathematical sciences, and mathematical physics can be formulated as VIP.We denote the solution set of a VIP by Ω. Due to the fruitful applications of the VIP, many researchers in this area have developed different iterative techniques to solve VIP (1.1).In particular, Goldsten in [20] introduced an iterative technique defined as follows: for all n ∈ N, where λ ∈ (0, 2α L 2 ), F is α-strongly monotone and L-Lipschitz continuous and P C is a metric projection defined from H onto C. The author established that the iterative method (1.2) converges to the solution set of VIP (1.1).However, it was observed that if F monotone and L-Lipschitzian continuous, the iterative method (1.2) may not converge to the solution set of VIP (1.1), see [22] and the reference therein for details.In addition, computing the value of λ may be very difficult or impossible.In the light 2 1 D. O. Peter, 1,2,3 A. A. Mebawondu, 4,5 G. C. UGWUNNADI, 1 P. Pillay and 1 O. K. Narain of these draw back, Korpelevich in [24] introduced and studied the Extragradient Method (EM) iterative technique defined as follows: (1.3) for all n ≥ 1, where λ n ∈ (0, 1 L ), F is monotone and L-Lipschitz continuous and P C is a metric projection defined from H onto C.This method was able to provide an affirmative answer to the question of weakening the cost operator, however, the computation of λ n remains a challenge.More so, another set back of this technique (1.3) is that it requires two projections onto the feasible set C per iteration, which is costly when C is not a simple structure.Since the inception of EM, many authors have introduced, modified and studied different EM in which the cost operator F is monotone and pseudomonotonicity.For example, He et at.[23], Apostol et al. [2], He et al. [22], Ceng et al. [4], Ceng et al [5], Nadezhkina and Takahashi [27] and many others.In addition, the notion of VIP (1.1) has also been extended and generalized by many authors.For example, Mainge in [28] introduced and studied the notion of Bilevel Variational Inequality Problems (BVIP).The BVIP is defined as follows: where G : H → H is L-Lispschitz continuous and γ-strongly monotone.It is easy to see that the BVIP (1.4) is a problem that is made up of the VIP (1.1) as a constraint.He proposed the following extragradient technique: where and α n ⊂ (0, 1) such that lim n→∞ α n = 0 and ∞ n=1 α n = +∞.It was established that the sequence generated by {x n } converges strongly to a unique solution of problem BVIP (1.4).It is easy to see that the iterative technique (1.5) has at least two setbacks, for example L , and the double metric projection (P C ).In order to overcome this setbacks, researchers have introduced the Tseng type iterative technique, the projection contraction iterative technique and the subgradient extragradient iterative technique that are self adaptive, see [35,37,38,39,42] and the reference therein for details.In particular, Tan et al., [38] introduced and studied the following iterative technique; for all n ∈ N, and for all n ∈ N, where F is L-Lipschitz continuous, pseudomonotone and sequentially weakly continuous, G is α-strongly monotone and L 1 -Lipschitz continuous.They established that the iterative techniques (1.2) and (1.7) converge strongly to a unique solution of the BVIP (1.4) using some standard assumptions.The question is still wide open, if some (all) of the mentioned iterative techniques can further be improved.One of the ways in which these iterative techniques have been improved over the years is the introduction of the inertial extrapolation technique.In 1964, Polyak in [32] introduced an inertial extrapolation as an acceleration technique process for solving the smooth convex minimization problem.Since then, this technique has been employed by research to improve their iterative techniques.The inertial technique requires the first two initial terms of the iterative technique and the next iterate is defined by making use of the previous two iterates.Since inception of the inertial extrapolation, many authors have modified, extended and generalized the technique, see [15,16,35] and the references therein.However, it has always been a question of interest, if the extrapolation technique can further be improved.The importance of the BVIP cannot be overemphasized.The BVIP (1.4) has been applied to different areas of mathematical sciences, engineering, physics and so on.For example,the BVIP (1.4) has applications in equilibrium constraints, bilevel convex programming models, minimum-norm problems with the solution set of variational inequalities, bilevel linear programming, image restoration and many more see ( [1,12,18,19,26,40,41]) and the references therein.Due to these applications, many authors have introduced different iterative techniques for solving the BVIP in the framework of Hilbert spaces (see, [1,15,16,17,28,29,39] and the references therein).It is well-known that the underlying cost operators have crucial roles to play in real applications of these iterative methods.In the light of this introducing an iterative technique with weaker cost operators and better rate of convergence is highly sorted after.Having consider the above discussed literatures and the references therein, it is natural to ask the following question: 1. Can we construct an efficient inertial type iterative technique that does require the knowledge of the Lipschitz constant during implementation of the algorithm?
2. Can we construct an iterative technique for a BVIP (1.4) in which the cost operators are quasimonotone and α-strongly monotone in the framework of infinite dimensional Hilbert spaces and obtain strong convergence?
3. Can we further explore the iterative techniques (1.7) and (1.5), in which the cost operator is quasimonotone and strongly monotone and the knowledge of the Lipschitz constant needed not to be known during implementation?
The purpose of this work is to provide and affirmative answer to the above questions by introducing a modified inertial iterative technique with self-adaptive step size for approximating the solution of quasimonotone BVIP (1.4).In addition, we use a modified inertial technique to accelerate the rate of convergence of our proposed methods.In addition, our numerical experiments justify that our method is better than other methods in the literature for solving the BVIP (1.4).The rest of this paper is organized as follows: In Section 2, we recall some useful definitions and results that are relevant for our study.In Section 3, we present our proposed method.In Section 4, we establish strong convergence of our method and in Section 5, we present some numerical experiments to show the efficiency and applicability of our method in the framework of infinite dimensional Hilbert spaces.Lastly in Section 6, we give the conclusion of the paper.

Preliminaries
In this section, we begin by recalling some known and useful results which are needed in the sequel.Let H be a real Hilbert space.The set of fixed points of a nonlinear mapping T : H → H will be denoted by F (T ), that is F (T ) = {x ∈ H : T x = x}.We denotes strong and weak convergence by "→" and "⇀", respectively.For any x, y ∈ H and α ∈ [0, 1], it is well-known that (2.1) (f) sequentially weakly continuous if for each sequence {x n }, we obtain {x n } converges weakly to x implies that T x n converges weakly to T x.
Let C be a nonempty, closed and convex subset of H.For any u ∈ H, there exists a unique point Lemma 2.4.[1] Let C be nonempty closed convex subset of a real Hilbert space H.For any x ∈ H and z ∈ C, we have Lemma 2.5.[1] Let H be a Hilbert space and F : H → H be a τ -strongly monotone and L-Lipschitz continuous operator on H. Let α ∈ (0, 1) and γ ∈ (0, 2τ L 2 ).Then for any nonexpansive operator T : H → H, we can associate for all x, y ∈ H, where ν = 1 − 1 − γ(2τ − γL 2 ) ∈ (0, 1).

Proposed Algorithm
In this section, we present our proposed method for solving a bilevel quasimonotone variational inequality problem.Assumption 1. Condition A. Suppose 1.The feasible sets C is nonempty set, closed and convex subsets of the real Hilbert space H..

2.
{S n } is a sequence of nonexpansive mapping on H.

Algorithm 1. Initialization Step:
Step 1: Choose x 0 , x 1 ∈ H, given the iterates x n−1 and x n for all n ∈ N, choose θ n such that 0 ≤ θ n ≤ θn , where with θ been a positive constant and {ǫ n } is a positive sequence such that ǫ n = •(β n ).

Proof
The proof that λ n+1 , is well defined follows similar approach as in Lemma 3.1 of [25], thus we omit it.Lemma 4.2.Let {x n } be a sequence generated by Algorithm 1 under Assumption 3.Then, {x n } is bounded.
for all n ∈ N. Then using Algorithm 1, we have Also, using Algorithm 1 and (2.2) It follows that In addition, using Algorithm 1, Lemma 2.5 and the fact that γ = 1 − 1 − α(2τ − αL 2 1 ) ∈ (0, 1), we have , then from the characteristic of the metric projection, we have which implies that which implies that Since λ n k > 0, we have Using our hypothesis and the fact that lim k→∞ λ n k > 0, , we have Now, observe that Since F is Lischitz continuous on H and our hypothesis, we have Now observe that Since, the subsequence {x n k } of {x n } is weakly convergent to a point x * ∈ H. Hence, using the fact that lim n→∞ w n km − y n km = 0, we have that {y n km } also converges to x * .Now passing the limit as m → ∞ in (4.19), we have Hence, x * ∈ Ω.
Secondly, we consider the case in which lim sup k→∞ F y n k , x − y n k = 0 for x ∈ C. Let {δ k } be a non-increasing positive sequence defined by By our hypothesis, it is easy to see that By our hypothesis and (4.22), we have for each k ≥ 1, since {y n k } ⊂ C, it implies that {F y n k } is strictly non-zero and lim inf k→∞ F y n k = N 0 > 0. We therefore deduce that In addition, let {ǫ n k } be a sequence defined by Combining (4.24) and (4.26), we have By quasimonotonicity of the operator F on H, we get that Now, observe that Combining (4.28), (4.29) and applying the well known Cauchy Schwartz inequality, we have Since F is Lipschitz continuous, we have Combining (4.25) and (4.33) and using the definition of {ǫ n k }, we have Since, the subsequence {x n k } of {x n } is weakly convergent to a point x * ∈ H. Hence, using the fact that lim n→∞ w n k − y n k = 0, we have that {y n k } also converges to x * .Taking limit as k → ∞, since δ k → 0, we have Proof Let p ∈ Φ, observe that for some N 2 > 0.
where Ψ n = θn γβ n x n − x n−1 N 2 + 2 α γ Gp, p − x n+1 .According to Lemma 2.3, to conclude our proof, it is sufficient to establish that lim sup k→∞ Ψ n ≤ 0 for every subsequence { x n k − p } of { x n − p } satisfying the condition: To establish that lim sup k→∞ Ψ n k ≤ 0, we suppose that for every subsequence It is easy to see from (4.37), Thus, we have lim k→∞ It is easy to see that, as k → ∞, we have x n k +1 ≤ 0. Thus, From Lemma 2.3, we have that lim n→∞ x n − p = 0.

Numerical Example
In this section, we will give some numerical examples which will show the applicability and the efficiency of our proposed iterative technique in comparison to Algorithm 1.7, and Algorithm 1.6.
Example 5.1.Let H = L 2 ([0, 1]) be equipped with the inner product It is easy to see that F is 1-Lipschitz continuous and monotone, and G τ -strongly monotone.We used this example due to Remark ?? so that we can compare.Let S n : L 2 ([0, 1]) → L 2 ([0, 1]) be defined by S n x(t) = sin x(t).

Figure 2 :
Figure 2: Example 5.2, Top Left: Case I; Top Right: Case II; Bottom Centered: case III.
.3) Since y n = P C (w n − λ n Aw n ) and p ∈ Φ ⊆ C, and by Lemma 2.4, we havey n − w n + λ n F w n , y n − p ≤ 0.It then follows that y n − w n , y n − p ≤ −λ n F w n , y n − p .Using the fact that y n ∈ C and p ∈ Φ, we have F y n , y n − p ≥ 0. Using the above facts and (3.4),we have (4.2) becomes