scholarly journals Strong Convergence of Extragradient-Type Method to Solve Pseudomonotone Variational Inequalities Problems

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 115 ◽  
Author(s):  
Nopparat Wairojjana ◽  
Nuttapol Pakkaranang ◽  
Habib ur Rehman ◽  
Nattawut Pholasa ◽  
Tiwabhorn Khanpanuk
Keyword(s):  
Variational Inequalities ◽  
Strong Convergence ◽  
Real Hilbert Space ◽  
Lipschitz Constant ◽  
Minimax Problems ◽  
Iterative Algorithms ◽  
Pseudomonotone Operators ◽  
Variable Stepsize ◽  
Type Method ◽  
Penalization Methods

A number of applications from mathematical programmings, such as minimax problems, penalization methods and fixed-point problems can be formulated as a variational inequality model. Most of the techniques used to solve such problems involve iterative algorithms, and that is why, in this paper, we introduce a new extragradient-like method to solve the problems of variational inequalities in real Hilbert space involving pseudomonotone operators. The method has a clear advantage because of a variable stepsize formula that is revised on each iteration based on the previous iterations. The key advantage of the method is that it works without the prior knowledge of the Lipschitz constant. Strong convergence of the method is proved under mild conditions. Several numerical experiments are reported to show the numerical behaviour of the method.

scholarly journals Iterative Design for the Common Solution of Monotone Inclusions and Variational Inequalities

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1504
Author(s):  
Li Wei ◽  
Xin-Wang Shen ◽  
Ravi P. Agarwal
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Iterative Design ◽  
Common Solution ◽  
Monotone Inclusions ◽  
New Ideas ◽  
The Common

Some new forward–backward multi-choice iterative algorithms with superposition perturbations are presented in a real Hilbert space for approximating common solution of monotone inclusions and variational inequalities. Some new ideas of constructing iterative elements can be found and strong convergence theorems are proved under mild restrictions, which extend and complement some already existing work.

scholarly journals New Inertial Forward-Backward Mid-Point Methods for Sum of Infinitely Many Accretive Mappings, Variational Inequalities, and Applications

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 466
Author(s):  
Li Wei ◽  
Yingzi Shang ◽  
Ravi P. Agarwal
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Metric Projection ◽  
Special Cases ◽  
Accretive Mappings ◽  
Number Sequences ◽  
Zero Points

Some new inertial forward-backward projection iterative algorithms are designed in a real Hilbert space. Under mild assumptions, some strong convergence theorems for common zero points of the sum of two kinds of infinitely many accretive mappings are proved. New projection sets are constructed which provide multiple choices of the iterative sequences. Some already existing iterative algorithms are demonstrated to be special cases of ours. Some inequalities of metric projection and real number sequences are widely used in the proof of the main results. The iterative algorithms have also been modified and extended from pure discussion on the sum of accretive mappings or pure study on variational inequalities to that for both, which complements the previous work. Moreover, the applications of the abstract results on nonlinear capillarity systems are exemplified.

scholarly journals Strong convergence inertial projection algorithm with self-adaptive step size rule for pseudomonotone variational inequalities in Hilbert spaces

2021 ◽  
Vol 54 (1) ◽  
pp. 110-128
Author(s):  
Nopparat Wairojjana ◽  
Nuttapol Pakkaranang ◽  
Nattawut Pholasa
Keyword(s):  
Variational Inequalities ◽  
Strong Convergence ◽  
Line Search ◽  
Real Hilbert Space ◽  
Lipschitz Constant ◽  
Projection Algorithm ◽  
Step Size ◽  
Computational Performance ◽  
Inertial Algorithm ◽  
Adaptive Step

Abstract In this paper, we introduce a new algorithm for solving pseudomonotone variational inequalities with a Lipschitz-type condition in a real Hilbert space. The algorithm is constructed around two algorithms: the subgradient extragradient algorithm and the inertial algorithm. The proposed algorithm uses a new step size rule based on local operator information rather than its Lipschitz constant or any other line search scheme and functions without any knowledge of the Lipschitz constant of an operator. The strong convergence of the algorithm is provided. To determine the computational performance of our algorithm, some numerical results are presented.

scholarly journals Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Songnian He ◽  
Caiping Yang
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Convex Functions ◽  
Level Sets ◽  
Variational Inequality Problem ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Nonlinear Problems ◽  
Original Domain

Consider the variational inequalityVI(C,F)of finding a pointx*∈Csatisfying the property〈Fx*,x-x*〉≥0, for allx∈C, whereCis the intersection of finite level sets of convex functions defined on a real Hilbert spaceHandF:H→His anL-Lipschitzian andη-strongly monotone operator. Relaxed and self-adaptive iterative algorithms are devised for computing the unique solution ofVI(C,F). Since our algorithm avoids calculating the projectionPC(calculatingPCby computing several sequences of projections onto half-spaces containing the original domainC) directly and has no need to know any information of the constantsLandη, the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.

scholarly journals Convergence theorems for composite viscosity approaches to systems variational inequalities in Banach spaces

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6267-6281
Author(s):  
Lu-Chuan Ceng ◽  
Jen-Chih Yao ◽  
Yonghong Yao
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Infinite Family ◽  
General System ◽  
Inequality Constraint ◽  
Implicit And Explicit

In this paper, we study a general system of variational inequalities with a hierarchical variational inequality constraint for an infinite family of nonexpansive mappings. We introduce general implicit and explicit iterative algorithms. We prove the strong convergence of the sequences generated by the proposed iterative algorithms to a solution of the studied problems.

scholarly journals Solving Generalized Mixed Equilibria, Variational Inequalities, and Constrained Convex Minimization

2014 ◽  
Vol 2014 ◽  
pp. 1-26 ◽  
Author(s):  
A. E. Al-Mazrooei ◽  
A. Latif ◽  
J. C. Yao
Keyword(s):  
Variational Inequalities ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Finite Family ◽  
Common Element ◽  
Monotone Mappings ◽  
Generalized Mixed Equilibrium ◽  
Fréchet Differentiable ◽  
Implicit And Explicit ◽  
Mixed Equilibria

We propose implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings in a real Hilbert space. We prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets under very mild conditions.

scholarly journals Strong Convergence Analysis of Iterative Algorithms for Solving Variational Inclusions and Fixed-Point Problems of Pseudocontractive Operators

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Zhangsong Yao ◽  
Yan-Kuen Wu ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Convergence Analysis ◽  
Iterative Methods ◽  
Iterative Algorithms ◽  
Fixed Point Problems ◽  
Variational Inclusions ◽  
Type Method ◽  
Common Solution ◽  
Mild Conditions

Iterative methods for solving variational inclusions and fixed-point problems have been considered and investigated by many scholars. In this paper, we use the Halpern-type method for finding a common solution of variational inclusions and fixed-point problems of pseudocontractive operators. We show that the proposed algorithm has strong convergence under some mild conditions.

scholarly journals Approximation Results for Variational Inequalities Involving Pseudomonotone Bifunction in Real Hilbert Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 182
Author(s):  
Kanikar Muangchoo ◽  
Nasser Aedh Alreshidi ◽  
Ioannis K. Argyros
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Hilbert Spaces ◽  
Optimization Problems ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Mathematical Problem ◽  
Lipschitz Constant ◽  
Step Size ◽  
Pseudomonotone Bifunction

In this paper, we introduce two novel extragradient-like methods to solve variational inequalities in a real Hilbert space. The variational inequality problem is a general mathematical problem in the sense that it unifies several mathematical models, such as optimization problems, Nash equilibrium models, fixed point problems, and saddle point problems. The designed methods are analogous to the two-step extragradient method that is used to solve variational inequality problems in real Hilbert spaces that have been previously established. The proposed iterative methods use a specific type of step size rule based on local operator information rather than its Lipschitz constant or any other line search procedure. Under mild conditions, such as the Lipschitz continuity and monotonicity of a bi-function (including pseudo-monotonicity), strong convergence results of the described methods are established. Finally, we provide many numerical experiments to demonstrate the performance and superiority of the designed methods.

scholarly journals Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Long He ◽  
Yun-Ling Cui ◽  
Lu-Chuan Ceng ◽  
Tu-Yan Zhao ◽  
Dan-Qiong Wang ◽  
...  
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Equilibrium Problem ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Common Solution ◽  
Implicit Rule

AbstractIn a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively.

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