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

Variational inequalities are considered the most signi ﬁ cant ﬁ eld in applied mathematics and optimization because of their massive and vast applications. The current study proposed a novel iterative scheme developed through a ﬁ xed-point scheme and formulation for solving variational inequalities. Modi ﬁ cation 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 signi ﬁ cant boost to the results.


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][3][4][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][7][8][9][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][16][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.

Formulation and Basic Results
A convex set is denoted by K in H (Hilbert space). We denote norm and inner by k·k and h·, · i, respectively. We consider a variational inequality: for general operator T, find y ∈ K such that 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].
For z ∈ H, y ∈ K holds for the inequality if and only if where P K is the projection of H onto Kðconvex setÞ.
It is also known that the P K is called projection operator, which is also nonexpansive and holds for the inequality.
Lemma 2. If y is a solution of VI (1), then y ∈ K satisfies the relation where ρ ≥ 0 is taken as constant and P K is considered the projection operator H onto K: From Lemma 2, it is obvious that y is a solution of VI (1), if and only if y satisfies the residue vector rðy, ρÞ defined by Related to the original inequality (1), we see the Wiener-Hopf equations (WHE) problem. To be more precise, let Q K = I − P K , where P K is the projection operator and I is the identity operator. For the operator T : H ⟶ H, then for finding z ∈ H, we have 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 Lemma 3 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.

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.

Journal of Function Spaces
We define the relation It is obvious that y ∈ K is a solution of the VI if and only if y ∈ K is satisfied with Equation (13).
Using (10) and (13), we can rewrite as This fact has motivated us to establish the new predictorcorrector self-adaptive iterative method for solving the VI (1).
The convergence of the newly established result of Algorithm 1 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 by H 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 fy n g be generated by Algorithm 1. If θ = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 − 2ρα + ρ 2 β 2 p ð1 + γβÞ < 1, then the sequences fy n g converges to y * , for Proof. Since y * is a solution of NVI (1), from Lemma 1, we have Applying Algorithm 1, from (19) and (24), we know that P K is nonexpansive: Since T is considered as strongly monotone and Lipschitz continuous with constant α and β: From (25), we have From (25) and (22), we get From (18) and (22), we get w n − w * k k= P K y n − γD y n , ρ n ð Þ− γTy n ½ − P K y * − γTy * ½ k k ≤ y n − γD y n , ρ n ð Þ− γTy n − y * + γTy * k k ≤ y n − y * − γD y n , ρ n ð Þ k k + γ Ty n − Ty * k k ≤ y n − y * − γD y n , ρ n ð Þ k k + γβ y n − y * k k: Consider y n − y * − γD y n , ρ n ð Þ k k 2 = y n − y * k k 2 − 2γ y n − y * , D y n , ρ n ð Þ h i + γ 2 D y n , ρ n ð Þ k k 2 :

Numerical Example
Example 1. We take the nonlinear complementarity problems: for finding y ∈ R n , we have Here, TðyÞ = D 1 ðyÞ + D 2 ðyÞ + q,we considerD 1 ðyÞas nonlinear part, andD 2 ðyÞ + qis taken as a linear part, and in ((33)), we take a special case of the VI (1). The matrix D 2 = B t B + C, where B is n × n matrix whose entries we generate randomly in the interval ð−5,+5Þ, and skew-symmetric matrix C is considered in the same way. The vector is denoted by q and is obtained in the interval ð−500,+500Þ. This is distributed uniformly. For easy problems, we take ð−500,+500Þ and ð−500,0Þ considered for the hard problem. In D 1 ðyÞ, the nonlinear part of TðyÞ, the components are D j ðyÞ = d j * arctan ðy j Þ, and d j is a random variable generated in ð0, 1Þ: For the output of the result, we consider, μ = 2/3, δ = 0:95, δ 0 = 0:95, ρ > 0 and γ = 1:95; the initial guess y 0 = ð0, 0, 0, ⋯, 0Þ T . The computation starts with ρ 0 = 1 and stops as soon as krðy n , ρ n Þk ≤ 10 −7 : MATLAB is used for all codes. Table 1 represents the outcomes of Algorithm 1.

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.

Data Availability
The manuscript included all required data and information for its implementation.

Conflicts of Interest
All authors declare no conflicts of interest in this paper.