Variational Inequality Problem Research Articles

Best proximity point results for generalized proximal $Z$-contraction mappings in metric spaces and some applications

In this paper, we define generalized proximal Z-contraction mappings of first and second kind in a metric space (X, d). The existence of best proximity point is shown for the defined mappings under some specific conditions which generalizes and extends some existing results of Olgun et al. [23] and Abbas et al. [1]. Suitable examples are given to justify the derived results. Some applications are also shown via fixed point formulation for such mappings in variational inequality problem and homotopy result.

Subgradient Extragradient Method for Finite Lipschitzian Demicontractions and Variational Inequality Problems in a Hilbert Space

In this research, the modified subgradient extragradient method and K-mapping generated by a finite family of finite Lipschitzian demicontractions are introduced. Then, a strong convergence theorem for finding a common element of the common fixed point set of finite Lipschitzian demicontraction mappings and the common solution set of variational inequality problems is established. Furthermore, numerical examples are given to support the main theorem.

Best proximity points for generalized (F, R)-proximal contractions

We present the notion of generalized (F, R)-proximal non-self contractions and prove best proximity point theorems in complete metric spaces endowed with an arbitrary binary relation. An example is given to vindicate our claims. We also show that the edge preserving structure is a particular case of the binary relation R. Moreover, an application to variational inequality problem is given in order to demonstrate the efficacy of our results.

Strong and Weak Convergence Theorems for the Split Feasibility Problem of (β,k)-Enriched Strict Pseudocontractive Mappings with an Application in Hilbert Spaces

The concept of symmetry has played a major role in Hilbert space setting owing to the structure of a complete inner product space. Subsequently, different studies pertaining to symmetry, including symmetric operators, have investigated real Hilbert spaces. In this paper, we study the solutions to multiple-set split feasibility problems for a pair of finite families of β-enriched, strictly pseudocontractive mappings in the setup of a real Hilbert space. In view of this, we constructed an iterative scheme that properly included these two mappings into the formula. Under this iterative scheme, an appropriate condition for the existence of solutions and strong and weak convergent results are presented. No sum condition is imposed on the countably finite family of the iteration parameters in obtaining our results unlike for several other results in this direction. In addition, we prove that a slight modification of our iterative scheme could be applied in studying hierarchical variational inequality problems in a real Hilbert space. Our results improve, extend and generalize several results currently existing in the literature.

Double inertial extragradient algorithms for solving variational inequality problems with convergence analysis

In this paper, we introduce a novel dual inertial Tseng's extragradient method for solving variational inequality problems in real Hilbert spaces, particularly those involving pseudomonotone and Lipschitz continuous operators. Our secondary method incorporates variable step‐size, updated at each iteration based on some previous iterates. A notable advantage of these algorithms is their ability to operate without prior knowledge of Lipschitz‐type constants and without the need for any line‐search procedure. We establish the convergence theorem of the proposed algorithms under mild assumptions. To illustrate the numerical behavior of the algorithms and to make comparisons with other methods, we conduct several numerical experiments. The results of these evaluations are showcased and thoroughly examined to exemplify the practical significance and effectiveness of the proposed methods.

Strong convergence theorem of a new modified Bregman extragradient method to solve fixed point problems and variational inequality problems in general reflexive Banach spaces

In this paper, we introduce a new modified self-adaptive extragradient method with the inertial technique for solving the variational inequality problem of a pseudomonotone mapping and the fixed point problem of a Bregman relatively nonexpansive operator in a general reflexive Banach space. We prove a strong convergence theorem of our algorithm under some suitable assumptions. Finally, some numerical experiments are provided to demonstrate the effectiveness of the suggested iterative method.

Responding to consumer environmental awareness: A variational inequality approach to green manufacturing optimization

In response to increasing consumer environmental awareness, our study introduces a sustainable supply chain optimization model that stimulates competition among multiple green manufacturers within an uncertain market context. Each manufacturer crafts a selection of products showcasing various degrees of greenness, purposefully designed to meet the expanding demand of environmentally conscious consumers. This strategic model emphasizes profit maximization for green manufacturers, necessitating the definition of ideal greenness levels and production quantities for an array of product types. By employing variational inequality theory, we transmute the preliminary optimization model into an interconnected variational inequality problem. We propose an improved Euler algorithm, a proficient tool for attaining equilibrium solutions. Empirical scenarios inspired by the electric vehicle industry lend credibility to our model, indicating that the surge in consumer environmental awareness compels manufacturers to diversify their offerings and accentuate their products’ greenness levels. Intriguingly, the research suggests that escalating competition can incentivize manufacturers to elevate their products’ greenness levels, strategically countering potential profit erosion. In addition, the study reveals that intensifying consumers’ green utility sensitivity doesn’t invariably augment demand for manufacturers, even with significant improvements in product greenness. This paradox highlights the necessity for manufacturers to strategically adjust their production approaches in light of varying consumer preferences. Moreover, when the level of consumer environmental awareness achieves a certain level, the fierce competition might not always be harmful to some manufacturers and the industry if they can adopt effective strategies to improve green utility sensitivity.

Golden ratio method for solving monotone variational inequality problems in Hadamard spaces

In this article, a generalized variational inequality problem in the setting of Hadamard spaces is introduced and analyzed. For approximating a solution of the problem when the underlined mapping is monotone, an adaptive algorithm that requires the computation of only one proximal operator at each iteration is proposed. The work gives a step towards investigating monotone variational inequality problems in geodesic settings.

Strongly convergent inertial projection and contraction methods for split variational inequality problem

Strongly convergent inertial projection and contraction methods for split variational inequality problem

An inertial method for solving split equality quasimonotone Minty variational inequality problems in reflexive Banach spaces

An inertial method for solving split equality quasimonotone Minty variational inequality problems in reflexive Banach spaces

A modified inertial Tseng technique of Bilevel variational inequality problem with application to image processing

A modified inertial Tseng technique of Bilevel variational inequality problem with application to image processing

Tseng’s extragradient method with double projection for solving pseudomonotone variational inequality problems in Hilbert spaces

Tseng’s extragradient method with double projection for solving pseudomonotone variational inequality problems in Hilbert spaces

Strong convergence theorem for split variational inequality problems involving non-lipschitizian pseudomonotone and finite family of $$\eta $$-demimetric operators

Strong convergence theorem for split variational inequality problems involving non-lipschitizian pseudomonotone and finite family of $$\eta $$-demimetric operators

Finite termination of the optimal solution sequence in parametric optimization

This paper investigates the finite termination of the optimal solution sequence in parametric optimization as a subproblem of the equilibrium problems. By introducing an augmented mapping on the solution set of the equilibrium problems, we establish the concept of augmented weak sharpness for this set relative to the sequence of optimal solutions in parametric optimization. Augmented weak sharpness is an extension of the concepts of weak sharpness and strongly non-degeneracy. Under the condition that the solution set of the equilibrium problems is augmented weak sharpness, sufficient conditions for the finite termination of the sequence of optimal solutions in parametric optimization are provided. For some special cases of the equilibrium problems (Mathematical Programming (MP), Variational Inequality Problems (VIP), etc.), these results extend the corresponding results under weak sharpness or strongly non-degeneracy conditions in the existing literature. Additionally, they also provide weaker sufficient conditions for the finite termination of many optimization algorithms.

On the intermixed method for mixed variational inequality problems: another look and some corrections

We explore the intermixed method for finding a common element of the intersection of the solution set of a mixed variational inequality and the fixed point set of a nonexpansive mapping. We point out that Khuangsatung and Kangtunyakarn’s statement [J. Inequal. Appl. 2023:1, 2023] regarding the resolvent utilized in their paper is not correct. To resolve this gap, we employ the epigraphical projection and the product space approach. In particular, we obtain a strong convergence theorem with a weaker assumption.

Strong convergence of split equality variational inequality, variational inclusion, and multiple sets fixed point problems in Hilbert spaces with application

This paper introduces an innovative inertial simultaneous cyclic iterative algorithm designed to address a range of mathematical problems within the realm of split equality variational inequalities. Specifically, the algorithm accommodates finite families of split equality variational inequality problems, infinite families of split equality variational inclusion problems, and multiple-sets split equality fixed point problems involving demicontractive operators in infinite-dimensional Hilbert spaces. The algorithm integrates well-established methods, including the cyclic method, the inertial method, the viscosity approximation method, and the projection method. We establish the strong convergence of this proposed algorithm, demonstrating its applicability in various scenarios and unifying disparate findings from existing literature. Additionally, a numerical example is presented to validate the primary convergence theorem.

Two-step inertial Tseng’s extragradient method for solving quasimonotone variational inequalities

We focus on quasimonotone variational inequality problems with non-Lipschitz operator and combine some operator theory techniques to establish a weak convergent fixed point iterative method to approximate the solution of the presented problems. The convergence analysis assumes very mild conditions and strictly weaker assumption on the cost operator associated with the problems and thus generalizes and extend recent related results in the literature. Finally, several numerical examples illustrate the potential and advantages of the proposed algorithm over some existing methods are presented.

Weighted variational inequalities on Hadamard manifolds

In this study, we extend the weighted variational inequality problem (WVIP) and the Minty weighted variational inequality problem (MWVIP) from linear spaces to nonlinear spaces, specifically Hadamard manifolds, and show the relationships between WVIP and vector variational inequalities. In addition, we establish the concept of weighted monotonicities on the Hadamard manifold and prove the existence of many solutions.

Modified inertial viscosity extrapolation method for solving quasi-monotone variational inequality and fixed point problems in real Hilbert spaces

In this paper, we introduce and study a viscous-type extrapolation algorithm for finding a solution of the variational inequality problem and a fixed point constraint of quasi-nonexpansive mappings under the scope of real Hilbert spaces when the underlying cost operator is quasi-monotone. The method involves inertial viscosity approximation and a constructed self-adjustable step size condition that depends solely on the information of the previous step. We establish a strong convergence result of the proposed method under certain mild conditions on the algorithm parameters. Finally, to demonstrate the gain of our method, some numerical examples are presented in comparison with some related methods in literature.

Inertial Algorithms for Solving Nonmonotone Variational Inequality Problems

Inertial Algorithms for Solving Nonmonotone Variational Inequality Problems