# Self-Adaptive Predictor-Corrector Approach for General Variational Inequalities Using a Fixed-Point Formulation

Authors: Kubra Sanaullah; Saleem Ullah; Muhammad Shoaib Arif; Kamaleldin Abodayeh; Rabia Fayyaz
Journal: Journal of Function Spaces
 (2022)

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## Abstract

A literature review revealed that the general variational inequalities, fixed-point problems, and Winner–Hopf equations are equivalent. In this study, general variational inequality and fixed-point problem are considered. We introduced a new iterative method based on a self-adaptive predictor-corrector approach for finding a solution to the GVI. Adaptations in the fixed-point formulation and self-adaptive techniques have been used to predict a novel iterative approach. Convergence analyses of the suggested algorithm are demonstrated. Moreover, numerical analysis shows that we establish the new best method for solving general variational inequality which performs better than the previous one. Furthermore, it is known that GVI consisted of several classes including variational inequalities and related optimization problems, and results obtained in this study continue to hold for these problems.


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## Body

## 1. Introduction

Applied mathematics has adorned the most promising and panoramic field referred to as variational inequality theory. This theory is a powerful unifying methodology for the study of equilibrium problems and provides us algorithms with accompanying convergence analysis for computational purposes. Therefore, in recent few years, various branches of mathematical and engineering sciences can be transformed in the framework of variational inequalities such as electronics, heat transportation, elasticity, optimization, network analysis, water resources, game theory, equilibrium problems in economics, mechanics, and traffic analysis; see [1–10]. Such remarkable development claims the most simple and unidirectional models of linear and nonlinear techniques. The idea of variational inequalities was first originated by Stampacchia [11]. Related to the variational inequalities, we have the concept of the Wiener–Hopf equations and general variational inequalities which were introduced by Noor [12] and Shi [13] in conjunction with variational inequalities from different points of view.Several conventional improvements approach to establish the solutions for open, moving boundary value problems and asymmetric obstacle, unilateral, even-order, and odd-order problems utilizing general variational inequalities, see [13–18]. Equivalent effects of general variational inequalities and fixed-point problems utilizing the projection techniques in recent days are an active research field, see [8, 19–21]. Quantitative knowledge of pseudocontractive and nonlinear monotone (accretive) operators combined with Lipschitz-type conditions is vital to prove the convergence of fixed-point iterative procedures. The phenomena of variational inequalities have a significant contribution to solving the Wiener–Hopf equations. Salient features of Wiener–Hopf equations and optimization problems in the presence of variational inequalities are addressed by Shi, see [12, 20–23].Variants of projection methods such as Wiener–Hopf equation techniques, auxiliary principle scheme, decomposition, and dynamical systems are advanced for solving various kinds of variational inequalities and other related optimization problems, see [8, 17–19, 22–26]. A detailed study by Lions and Stampacchia revealed the utilization of such tools for finding the detailed solution of variational inequalities was in consumption a long time ago, see [27, 28]. The primary objective for employing these abilities is to keep the variational inequality and the fixed-point problem similar through projection. Based on this formula, many projection methods for resolving variational inequalities can be developed. This approach has been critical. Convergence of the projection method has a drawback; it required a strongly monotone and Lipschitz continuous operator, which has limited much application. Therefore, innovative methods or modifications in the projection method are required to diversify the field.Publications such as [23, 29–36] comprised of extra gradient-type methods which delimited the projection phenomenon by taking additional forward steps, and projection at each iteration is considered according to the double projection. These methods are a predictor-corrector tool and have been suggested to quantify variational inequalities and their special cases. We improve the recent best results for GVI by introducing innovative iterative methods.The self-adaptive predictor-corrector approach is the primary goal of this research, which modifies the fixed point by incorporating a generalized residue vector that includes general variational inequalities. For the convergence of the method, we require only pseudomonotonicity, which is considered a weaker condition than monotonicity. It is comprehended that the proposed model is simple and robust. We can see from the numerical results that the proposed technique is both rapid and easy to implement, as demonstrated by an example.


## 2. Preliminaries

We denote Hilbert spaceH with the norm and inner product by ⋅ and ,, respectively. A convex set is represented by M in H, and let A,ϕ:H⟶H be considered the nonlinear operators. For finding α∈H,ϕα∈M, such that(1)Aα,ϕβ−ϕα≥0,for allβ∈H:ϕβ∈M.Problem (1) is called the general variational inequalities(GVI), considered by Noor in 1988. We have observed that a large number of problems in pure and applied mathematics related to physical sciences, engineering, equilibrium, moving, nonsymmetric, unified, obstacle, and contact can be discussed and studied via inequalities (1), see [18, 19, 22].Forϕ≡I (take identity operator), problem (1) reduces to finding α∈M, such that(2)Aα,β−α≥0,∀β∈M.Problem (2) is defined by the original variational inequalities introduced by Stampacchia, see [11].The following concept and known results are required to approach the main algorithms.Definition 1.
LetA:H⟶H be called the operator and also ϕ-pseudomonotone if Aα,ϕβ−ϕα≥0 provides Aβ,ϕβ−ϕα≥0,∀α,β∈H.
It is considered [22, 24] that monotonicity implies pseudomonotonicity, but the converse does not exist.Lemma 1.
Forz∈H,α∈M holds for the inequality:(3)α−z,β−α≥0,∀β∈M,if and only if(4)α=PMz,where PM is called the projection of H onto the convex set M.It is also known thatPM is called the projection operator and also nonexpansive which satisfies the following inequality:(5)PMz−α≤z−α−z−PMz.Lemma 2.
α is a solution of the given GVI (1) if and only if α∈M satisfies the relation(6)ϕα=PMϕα−ρAα,where ρ≥0 is taken as the constant and PM is considered the projection operator H onto M.Residue vectorR1α is defined by(7)R1α≔ϕα−PMϕα−ρAα.From Lemma1, we can see that α satisfy (1) if and only if α is a zero point of the function:(8)R1α≔0.For the GVI (1), we consider the problem for the Wiener–Hopf equations. Let QM=I−PM, where I is the identity operator and PM is projection operation. For the operators A,ϕ:H⟶H, and ϕ−1 exists; then, for finding z∈H, we have(9)ρAϕ−1PMz+QMz=0,where (9) is the general Wiener–Hopf equation(GWHE), investigated by Noor [18]. We have seen that the Wiener–Hopf equations are considered and used to establish various efficient and powerful iterative schemes.Lemma 3.
The functionα∈H:ϕα∈M satisfies inequalities (1) if and only if z∈H satisfies equation (9), provided(10)ϕα=PMz,(11)z=ϕα−ρAα.Lemma3 provides that the GVI (1) is equivalent to GWHE (9). This fixed-point formulation was considered by Noor [24] to establish various iterative schemes for solving the GVI and other optimization theory and related problems.This useful scheme has been considered to make and establish a self-adaptive method for solving the GVI (1).By using (7) and (10) and (11), the GWHE (9) can be modified in the form:(12)0=ϕα−PMϕα−ρAα−ρAα+ρAϕ−1PMϕα−ρAα,=R1α−ρAα+ρAϕ−1PMϕα−ρAα.Forω∈0,1, (9) can be mentioned as(13)ϕα=PMϕα−ωd1α,where(14)d1α=R1α−ρAα+ρAϕ−1PMϕα−ρAα.This equivalent modification has been considered by Noor [31] for solving the general variational inequalities (GVI).Algorithm 1.
Step 0. We set the parameters as follows. For αo∈H, set n=0, and take δ0,δ∈0,1,ϵ>0,γ∈1,2,μ∈0,1, and ρ>0.Step 1. If R1αn<ϵ, then we terminate; otherwise, take ρn=ρμmn, where mn finds the smallest nonnegative integer that satisfies the inequality ρnAαn−Aϕ−1PMϕαn−ρnAαn,R1αn≤δR1αn2.Step 2. Compute d1αn=R1αn−ρnAαn+ρnAϕ−1PMϕαn−ρnAαn and ωn=.1−δR1αn2/d1αn2.Step 3. Find the next iteration, ϕαn+1=ϕαn−ωnd1αn.Step 4. If ρnAαn−Aϕ−1PMϕαn−ρnAαn,R1αn≤δ0R1αn2, then again take ρ=ρn/μ, else ρ=ρn. Consider n=n+1, and start iteration from step 1.


## 3. Main Results

We suggest predictor-corrector techniques for updating the scheme to find the solution of the GVI (1):(15)ϕy=PMϕα−ρAα,ϕα=PMϕy−ρAy.Here, we suggest the residue vector involving projection by the relation:(16)Rα=ϕα−PMϕy−ρAy=ϕα−ϕy=ϕα−PMPMϕα−ρAα−ρAϕ−1PMϕα−ρAα.It is clear thatα∈H,andϕα∈M is a solution of the GVI (1) if and only if α∈H,andϕα∈M is satisfied with the residue vector:(17)Rα=0.Since the convex set is defined byM, then, for all η∈0,1, using (16), we have(18)ϕα=PMϕα−ωηRα+ρAϕ−1ϕα−ηRα,=PMϕα−ωdα.where(19)dα=ηRα+ρAϕ−1ϕα−ηRα.We now analyze and recommend the following predictor-corrector scheme for finding the GVI (1).Algorithm 2.
Step 0. For parameters ϵ>0,ρ>0,δ0,δ∈0,1,γ∈0,1,μ∈0,1, and αo∈H, we start from n=0,η∈0,1.Step 1. We again take ρn=ρ. If Rαn<ϵ, then computation stops; otherwise, we continue and consider ρn=ρμmn and find the smallest nonnegative integer mn, which satisfies the inequality ρnAαn−Aϕ−1ϕαn−ηnRαn,Rαn≤σRαn2,σ∈0,1.Step 2. Calculate the next iterate:(20)ϕαn+1=PMϕαn−ωndαn,where(21)dαn=Rαn+ρnAϕ−1ϕαn−ηnRαn,ϕαn+1=PMϕαn−ωndαn,(22)ωnRαn,Dαndαn2.Step 3. If(23)ρnAαn−Aϕ−1ϕαn−ηnRαn,Rαn≤σRαn2,andσ∈0,1,then again we setρ=ρn/μ, else by setting ρ=ρn. Take n=n+1, and repeat step 1.
Here,ωn is taken as corrector step size which contains the GWHE (9).
We consider the convergence of the main established results, which is the main target of this research.Theorem 1.
Ifα∗ is a solution of inequality(1) and the operator A:H⟶H is ϕ-pseudomonotone, then(24)ϕα−ϕα∗,dα≥η−σRα2,∀α∈H.Proof.
Letα∗∈H be a solution of GVI (1). Then,(25)Aα∗,ϕβ−ϕα∗≥0,∀ϕβ∈M,since T is ϕ-pseudomonotone. Taking ϕβ=ϕα−ηRα in (25), we have(26)Aϕ−1ϕα−ηRα,ϕα−ηRα−ϕα∗≥0.
This implies that(27)Aϕ−1ϕα−ηRα,ϕα−ϕα∗≥ηAϕ−1ϕα−ηRα,Rα.
Takingz=ϕα−ρAα,α=PMϕy−ρAα, and β=ϕα∗ in (3), we obtain(28)0≤PMϕy−ρAα−ϕα+ρAα,ϕα∗−PMϕy−ρAα=ρAα−ϕα−PMϕy−ρAα,ϕα∗−ϕα+ϕα−PMϕy−ρAα=ρAα−Rα,ϕα∗−ϕα+Rα.
Using3.1,(29)=Rα−ρAα,ϕα−ϕα∗−Rα=Rα,ϕα−ϕα∗−ρAα,ϕα−ϕα∗−Rα−ρAα,Rα≤Rα,ϕα−ϕα∗−Rα−ρAα,Rα,from which, we have(30)Rα,ϕα−ϕα∗≥Rα−ρAα,Rα.
Adding (27) and (30), we obtain(31)ϕα−ϕα∗,Aϕ−1ϕα−ηRα+Rα≥ηRα−ρAα+Aϕ−1ϕα−ηRα,Rα.
Using (19), (21), (23), (27), and (31), we have(32)ϕα−ϕα∗,dα≥ηRα,Dα=ηRα,Rα−ηρRα,Aα−Aϕ−1ϕα−ηRα=ηRα2−ηρRα,Aα−Aϕ−1ϕα−ηRα≥η−σRα2,which is the desired result.Theorem 2.
Ifα∗∈H is a solution of GVI (1) and αn+1 is the approximate solution found from Algorithm 2, then(33)ϕαn+1−ϕα∗≤ϕαn−ϕα∗−ηn−σRαn4dαn2.Proof.
From (20), (21), (22), and (32), we have(34)ϕαn+1−ϕα∗2=PMϕαn−ωndαn−ϕα∗2≤ϕαn−ωndαn−ϕα∗2≤ϕαn−ϕα∗2−2ωnηn−σRαn2+ωn2dαn2=ϕαn−ϕα∗2−2ηn−σRαn,Dαndαn2Rαn2+Rαn,Dαn2dαn4dαn2≤ϕαn−ϕα∗2−2ηn−σ2Rαn4dαn2+ηn−σ2Rαn4dαn2=ϕαn−ϕα∗2−ηn−σ2Rαn4dαn2,which is the required result.Theorem 3.
Letαn+1 be the approximated solution obtained from Algorithm 2 and α∗∈M be a solution of GVI (1). If H is a finite-dimensional space, then limαnn⟶∞=α∗.Proof.
Letα∗∈M be a solution of GVI (1). From (34), we get that the sequence αn is bounded; we have(35)ηn−σ2Rαn4dαn2≤ϕαn−ϕα∗2,which shows both expressions are going to be zero when n⟶∞ such as(36)limRαnn⟶∞=0and(37)limηnn⟶∞=0,which implies (36) holds. Let α∗ be taken as the cluster point of αn, and consider the subsequence αni of the sequence αn converge to point α∗. We know continuity of R holds; we have(38)Rα∗=limRαnn⟶∞=0,which provides that α∗ is a solution of GVI (1) by Theorem 3 and(39)ϕαn+1−ϕα∗2≤ϕαn−ϕα∗2.
Thus, the sequenceαn converges exactly one cluster point and the consequences, and we obtain(40)limϕαnn⟶∞=ϕα∗.
Sinceϕ is injective, it gives that limηnn⟶∞=α∗∈H, which satisfies the GVI (1).
Suppose that (37) holds and limηnn⟶∞=0. If (32) does not hold, then, by taking the value of ηn, we obtain(41)σRαn≤ηnρnAαn−Aϕ−1ϕαn−ηnRαn,Rαn.
Letα∗ be the cluster point of αn and let αni be the subsequence αni converge to α∗. We apply the limit in (41); then,(42)σRα∗2≤0,which gives Rα∗=0, that is, α∗∈H is a solution of inequality (1), and by Lemma 1, inequality (41) holds. By repeating the same process and arguments, we approach that limαnn⟶∞=α∗, the desired result.


## 4. Numerical Example

Problem 1.
This problem is relevant to inequality (1), with ϕα=Aα+q and Aα=α, where(43)A=4−20⋯0014−2⋯00014⋱00⋮⋮⋱⋱⋱⋮000⋱⋱−2000⋯14,q=1⋮⋮1.
We set the following domain for the considered problem:M=α∈Rn/0≤αi≤1,fori=1,2,3,…,n. Results for Algorithm 1 are mentioned in Table 1. Tables 2 and 3 represent the outcomes of Algorithm 2 with the initial point α0=−A−1q for the order n=100 of the generated matrix . For all output, we consider μ,δ∈0,1,γ∈1,2, and ρ>0. The iterative process will stop when we have rαn,ρn≤10−7.
From Tables1 and 2, we observe with the change of parameters that the number of iterations also varies. Table 2 gives the results for the newly established method (Algorithm 2). From the output, we observe that the newly established method converges more quickly than Algorithm 1 for solving the main GVI.
From Tables3 and 4, we see that, in the new iterative scheme, the number varies (iterations) by changing the parameters δ,ρ, and μ. By changing the parameters accordingly, we can reduce the number of iterations.Table 1
Algorithm1 (numerical results).
Parametersρ=7,δ=0.02,ρ=7,δ=0.02,ρ=7,δ=0.02,μ=0.7μ=0.8μ=0.9Iterations142247Table 2
Algorithm2 (numerical results).
Parametersρ=7,δ=0.02,ρ=7,δ=0.02,ρ=7,δ=0.02,μ=0.7μ=0.8μ=0.9Iterations121942Table 3
Algorithm2 (numerical results).
Parametersρ=5,δ=0.3,ρ=7,δ=0.2,ρ=7,δ=0.05,μ=0.6μ=0.4μ=0.7Iterations4311Table 4
Algorithm2 (numerical results).
Parametersρ=4,δ=0.2,ρ=4,δ=0.1,ρ=4,δ=0.04,μ=0.8μ=0.8μ=0.8Iterations91216


## 5. Conclusion

For this study, the predictor-corrector self-adaptive method has been applied and considered to find the solution of the GVI. We used pseudomonotone of the operator, which is considered as a weaker condition than monotonicity. We also proved the convergence analysis, which is the main motivation of this paper. It has been analyzed that the new technique is more efficient than the already proved methods. The efficiency of the method has been illustrated through an example. Comparison is provided with other known methods. The numerical results reflect the output of our newly established algorithms well for the considered problem.



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