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.
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 by in (Hilbert space). We denote norm and inner by and , respectively. We consider a variational inequality: for general operator , find 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 holds for the inequality
if and only if
where is the projection of onto .
It is also known that the is called projection operator, which is also nonexpansive and holds for the inequality.
Lemma 2. If is a solution of VI (1), then satisfies the relation where is taken as constant and is considered the projection operator onto
From Lemma 2, it is obvious that is a solution of VI (1), if and only if satisfies the residue vector defined by
Related to the original inequality (1), we see the Wiener-Hopf equations (WHE) problem. To be more precise, let , where is the projection operator and is the identity operator. For the operator , then for finding , 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 , if and only if 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.
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.
Using (6), (8), and (10), the WHE (7) can be written in the form
We define the relation
It is obvious that is a solution of the VI if and only if is satisfied with Equation (13).
Using (10) and (13), we can rewrite as
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 and set
Step 2: Set ; if then computation stops; otherwise, the iteration will continue to find the nonnegative integer, and take which satisfies the inequality
where
Step 3: Compute
where
Step 4: Get the next iterate
and then set , else set , and go to Step 2
We observe that Algorithm 1 is refinement and addition of the standard procedure. Here, we consider 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 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 and ; we take as strongly monotone, where is Lipschitz continuous mapping on a convex subset of Let be a solution of VI (1) and let the sequences be generated by Algorithm 1. If then the sequences converges to for
Proof. Since is a solution of NVI (1), from Lemma 1, we have
Applying Algorithm 1, from (19) and (24), we know that is nonexpansive:
Since is considered as strongly monotone and Lipschitz continuous with constant and From (25), we have
Consider
We use the definition of and we obtain
From (27) and (31), we get where since , thus from (32) and Algorithm 1 for an arbitrarily chosen and consider initial points and obtained from Algorithm 1,which converge strongly to
4. Numerical Example
Example 1. We take the nonlinear complementarity problems: for finding , we have Here, we consideras nonlinear part, andis taken as a linear part, and in ((33)), we take a special case of the VI (1). The matrix , where is matrix whose entries we generate randomly in the interval , and skew-symmetric matrix is considered in the same way. The vector is denoted by and is obtained in the interval . This is distributed uniformly. For easy problems, we take and considered for the hard problem. In the nonlinear part of the components are , and is a random variable generated in
For the output of the result, we consider, and ; the initial guess . The computation starts with and stops as soon as MATLAB is used for all codes. Table 1 represents the outcomes of Algorithm 1.
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.
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.
Acknowledgments
We are thankful to the anonymous referees for their valuable comments which helped us to improve the quality of the paper. The authors wish to express their gratitude to Prince Sultan University for facilitating the publication of this article through the research lab Theoretical and Applied Sciences Lab. The authors would like to acknowledge the support of Prince Sultan University for paying the article processing charges (APC) of this publication.