# A Self-Adaptive Technique for Solving Variational Inequalities: A New Approach to the Problem

Authors: Muhammad Bux; Saleem Ullah; Muhammad Shoaib Arif; Kamaleldin Abodayeh
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Variational inequalities are considered the most significant field in applied mathematics and optimization because of their massive and vast applications. The current study proposed a novel iterative scheme developed through a fixed-point scheme and formulation for solving variational inequalities. Modification is done by using the self-adaptive technique that provides the basis for predicting a new predictor-corrector self-adaptive for solving nonlinear variational inequalities. The motivation of the presented study is to provide a meaningful extension to existing knowledge through convergence at mild conditions. The numerical interpretation provided a significant boost to the results.


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## Body

## 1. Introduction

Earlier, most of the equilibrium-related queries were resolved by variational inequalities that are a mathematical theory. In this regard, Stampacchia [1] is considered a pioneer who initially introduced variational inequalities in 1964. At the end of 1964, Stampacchia extended his work by introducing partial differential equations. Since then, this field has become the most emerging and demanding with extensive applications in optimization and control, economics, movements, engineering sciences, and equilibrium problems. Massive utilization of variational inequalities in applied sciences made it branched and more generalized to interact with other fields [2–5], hence proved the novelty and productivity of variational inequalities. Most of the profound task for researchers is to work on extensions and generalized inequalities regarding their applications; consequently, it gives rise to pure and applied mathematics problems. Modifications in variational inequalities produced advances in numerical methods [6–10], sensitivity analysis, and the dynamical system that are efficient in solving mathematics-related problems. Theory and algorithmic advancements meet in the theory of variational inequalities, opening up a brand-new field of application [7, 11, 12]. These issues necessitate a combination of convex, functional, and numerical analysis techniques. There are numerous exciting applications for this fascinating section of applied mathematics in the fields of business, finance, economics, and the social, as well as the pure and applied sciences (see [3, 9, 13, 14] and the references therein for applications and numerical approaches). Such extraordinary progress is based on the most basic and unidirectional linear and nonlinear approaches.A fundamental problem associated with variational inequalities is the establishment of fast numerical methods. A projection-type method and its variant solve many optimization problems and are also related to variational inequalities. Variational inequalities and fixed-point issues with equivalent effects utilizing projection techniques have grown in popularity in recent years as a study focus. To prove the convergence of fixed-point iterative methods, quantitative knowledge of pseudocontractive and nonlinear monotone (accretive) operators combined with Lipschitz type conditions is required (see [15–17]). 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 [17]. Together with the Wiener-Hopf equation, the projection method is considered an important technique for approximating the solution of variational inequality problems. Constructing an equivalence between fixed-point problems and variational inequalities is made easier with the concept of the projection method. Utilizing variational problems, several conventional improved ways to establish solutions for open, moving boundary value problems, asymmetric obstacle, unilateral, even-order, and odd-order problems could be developed (see [4–7, 11, 15] and the references therein). An investigation into a new predictor-corrector self-adaptive strategy for solving nonlinear variational inequalities under known assumptions is suggested in the proposed study. It was possible to arrive at this fixed-point formulation using projection, variational inequalities, and Wiener-Hopf equations. Additionally, the convergence of the proposed method is discussed.


## 2. Formulation and Basic Results

A convex set is denoted byK in H (Hilbert space). We denote norm and inner by · and ·,·, respectively. We consider a variational inequality: for general operator T, find y∈K such that
(1)Ty,x−y≥0,∀x∈K.The inequality (1) is called the variational inequality (VI) introduced by Stampacchia [1]. A large number of problems related to equilibrium, nonsymmetric, physical sciences, engineering, moving boundary value problem, unified, obstacle, unilateral contact, and applied sciences can be discussed via the inequalities (1) [1, 6, 7, 12, 13].Lemma 1.
[13].
Forz∈H,y∈K holds for the inequality
(2)y−z,x−y≥0,∀x∈K,if and only if
(3)y=PKz,where PK is the projection of H onto Kconvexset.It is also known that thePK is called projection operator, which is also nonexpansive and holds for the inequality.
(4)PKz−y≤z−y−z−PKz.Lemma 2.
Ify is a solution of VI (1), then y∈K satisfies the relation
(5)y=PKy−ρTy,where ρ≥0 is taken as constant and PK is considered the projection operator H onto K.From Lemma2, it is obvious that y is a solution of VI (1), if and only if y satisfies the residue vector ry,ρ defined by
(6)ry,ρ=y−PKy−ρTy.Related to the original inequality (1), we see the Wiener-Hopf equations (WHE) problem. To be more precise, let QK=I−PK, where PK is the projection operator and I is the identity operator. For the operator T:H⟶H, then for finding z∈H, we have
(7)ρTPKz+QKz=0.Here, Equation (7) is the Wiener-Hopf equation (WHE), investigated by Shi [17]. This WHE (7) is considered more general and gives a unified framework to establish the various powerful and efficient iterative methods and numerical techniques (for the application of the WHE (7), see [17, 18]).Lemma 3.
The inequality (1) has a unique solution y∈K, if and only if z∈H satisfies the WHE (7), provided
(8)y=PKz,(9)z=y−ρTy.Lemma3 implies that the VI (1) is equivalent to WHE (7). Noor et al. [8, 18] considers this fixed-point formulation to establish various iterative schemes for solving the VI and other optimization and related problems.


## 3. Main Results and Algorithm

To solve the variational inequality (1), we will use an iterative approach that we are developing in this study. The relevant results, algorithm, and theory will be established to make an iterative process for solving the inequality. The convergence of the new technique will also be provided.We use the fixed-point formulation and suggest a predictor-corrector technique for upgrading the solution for VI.(10)w=PKy−γTy,forγ>0,(11)y=PKw−ρTw=PKPKy−γTy−ρTPKy−γTy.Using (6), (8), and (10), the WHE (7) can be written in the form
(12)0=y−PKy−ρTy−ρTy+ρTPKy−ρTy=ry,ρ−ρTy+ρTPKy−ρTy.We define the relation(13)Dy,ρ=ry,ρ−ρTy+ρTPKy−ρTy.It is obvious thaty∈K is a solution of the VI if and only if y∈K is satisfied with Equation (13).
(14)Dy,ρ=0.Using (10) and (13), we can rewrite as
(15)w=PKy−γDy,ρ−γTy,This fact has motivated us to establish the new predictor-corrector self-adaptive iterative method for solving the VI (1).Algorithm 1.
Step 1: Give∈>0,γ>0,δ∈0,1,δ0∈0,1,μ∈0,1,ρ>0, and y∗∈H set n=0
Step 2: Setρn=ρ; if ryn,ρ<ϵ, then computation stops; otherwise, the iteration will continue to find the mn nonnegative integer, and take ρn=ρμmn which satisfies the inequality
(16)ρnTyn−Twn≤δryn,ρn,where
(17)wn=PKyn−γnDyn,ρn−γnTyn
Step 3: Compute(18)dyn,ρn=ryn,ρn−ρnTyn+ρnTPKyn−ρnTyn,where
(19)ryn,ρn=yn−PKyn−ρTyn
Step 4: Get the next iterate(20)wn=PKyn−γDyn,ρn−γTyn,(21)yn+1=PKwn−ρTwn,
and then setρ=ρn/μ, else set ρ=ρn.n=n+1, and go to Step 2We observe that Algorithm1 is refinement and addition of the standard procedure. Here, we consider −γDyn,ρn−γTyn, the self-adaptive technique, or we can say the step-size. This technique and procedure are closely related to the projection residue.The convergence of the newly established result of Algorithm1 is the important part to consider under some suitable and mild conditions, which is the paper’s main target and motivation.Theorem 4.
Let real Hilbert space be denoted byH and T:K⟶H; we take α as strongly monotone, where β is Lipschitz continuous mapping on a convex subset K of H. Let y∗∈K be a solution of VI (1) and let the sequences ynbe generated by Algorithm 1. If θ=1−2ρα+ρ2β21+γβ<1, then the sequences yn converges to y∗, for
(22)0<ρ<2αβ2.Proof.
Sincey∗ is a solution of NVI (1), from Lemma 1, we have
(23)w∗=PKy∗−y∗Ty∗,forγ>0,(24)y∗=PKw∗−ρTw∗,forρ>0.Applying Algorithm1, from (19) and (24), we know that PK is nonexpansive:
(25)yn+1−y∗=PKwn−ρTwn−PKw∗−ρTw∗≤wn−w∗−ρTwn+ρTw∗.SinceT is considered as strongly monotone and Lipschitz continuous with constant α and β. From (25), we have
(26)wn−w∗−ρTwn−Tw∗2=wn−w∗2−2ρTwn−Tw∗,wn−w∗+ρ2Twn−Tw∗2≤wn−w∗2−2ραwn−w∗2+ρ2β2wn−w∗2=1−2ρα+ρ2β2wn−w∗2.From (25) and (22), we get
(27)yn+1−y∗≤1−2ρα+ρ2β2wn−w∗.From (18) and (22), we get
(28)wn−w∗=PKyn−γDyn,ρn−γTyn−PKy∗−γTy∗≤yn−γDyn,ρn−γTyn−y∗+γTy∗≤yn−y∗−γDyn,ρn+γTyn−Ty∗≤yn−y∗−γDyn,ρn+γβyn−y∗.Consider(29)yn−y∗−γDyn,ρn2=yn−y∗2−2γyn−y∗,Dyn,ρn+γ2Dyn,ρn2.We use the definition ofDyn,ρn, and we obtain
(30)yn−y∗−γDyn,ρn≤yn−y∗.From (28) and (30), we have
(31)wn−w∗≤1+γβyn−y∗.From (27) and (31), we get
(32)yn+1−y∗≤1−2ρα+ρ2β21+γβyn−y∗=θyn−y∗,where θ=1−2ρα+ρ2β21+γβ, since 0<θ<,∑n=0∞θn=∞, thus from (32) and Algorithm 1 for an arbitrarily chosen and consider initial points y0 and yn obtained from Algorithm 1,which converge strongly to y∗.


## 4. Numerical Example

Example 1.
We take the nonlinear complementarity problems: for findingy∈Rn, we have
(33)y≥0,Ty≥0,y,Ty=0.
Here,Ty=D1y+D2y+q,we considerD1yas nonlinear part, andD2y+qis taken as a linear part, and in ((33)), we take a special case of the VI (1). The matrix D2=BtB+C, where B is n×n matrix whose entries we generate randomly in the interval −5,+5, and skew-symmetric matrix C is considered in the same way. The vector is denoted by q and is obtained in the interval −500,+500. This is distributed uniformly. For easy problems, we take −500,+500 and −500,0 considered for the hard problem. In D1y, the nonlinear part of Ty, the components are Djy=dj∗arctanyj, and dj is a random variable generated in 0,1.For the output of the result, we consider,μ=2/3,δ=0.95,δ0=0.95,ρ>0 and γ=1.95; the initial guess y0=0,0,0,⋯,0T. The computation starts with ρ0=1 and stops as soon as ryn,ρn≤10−7. MATLAB is used for all codes. Table 1 represents the outcomes of Algorithm 1.Table 1
For Algorithm 1 (numerical results).
Numerical resultsNumerical resultsOrder of matrixAlgorithm1[8]nNo. It.No. It.10044442005555300484850031317004343


## 5. Conclusion

We have considered the new technique for solving inequality (1). We have applied the self-adaptive technique to control the step size under some mild conditions. Results have been compared with the published paper. It has been observed that the number of iterations is reduced by applying the new suggested method. This is an extension of the previously known results. This work can be enhanced further when the operator is pseudomonotone which is considered a weaker condition when the operator is strongly monotonicity. The numerical results reflect the output of our newly established algorithms well for the considered problems.



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