A Class of Novel Mann-Type Subgradient Extragradient Algorithms for Solving Quasimonotone Variational Inequalities

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1108
Author(s):  
Nopparat Wairojjana ◽  
Ioannis K. Argyros ◽  
Meshal Shutaywi ◽  
Wejdan Deebani ◽  
Christopher I. Argyros
Keyword(s):  
Variational Inequalities ◽  
Quantum Physics ◽  
Real Hilbert Space ◽  
Lipschitz Constant ◽  
Step Size ◽  
Infinite Dimensional ◽  
Iterative Schemes ◽  
Symmetry Principles ◽  
Line Search Method ◽  
Quasimonotone Operators

Symmetries play an important role in the dynamics of physical systems. As an example, quantum physics and microworld are the basis of symmetry principles. These problems are reduced to solving inequalities in general. That is why in this article, we study the numerical approximation of solutions to variational inequality problems involving quasimonotone operators in an infinite-dimensional real Hilbert space. We prove that the iterative sequences generated by the proposed iterative schemes for solving variational inequalities with quasimonotone mapping converge strongly to some solution. The main advantage of the proposed iterative schemes is that they use a monotone and non-monotone step size rule based on operator knowledge rather than a Lipschitz constant or some line search method. We present a number of numerical experiments for the proposed algorithms.

scholarly journals Halpern subgradient extragradient algorithm for solving quasimonotone variational inequality problems

2021 ◽  
Vol 38 (1) ◽  
pp. 249-262
Author(s):  
PONGSAKORN YOTKAEW ◽  
◽  
HABIB UR REHMAN ◽  
BANCHA PANYANAK ◽  
NUTTAPOL PAKKARANANG ◽  
...  
Keyword(s):  
Variational Inequalities ◽  
Hilbert Spaces ◽  
Line Search ◽  
Lipschitz Constant ◽  
Iterative Sequence ◽  
Step Size ◽  
Infinite Dimensional ◽  
Extragradient Algorithm ◽  
Line Search Method ◽  
Quasimonotone Operators

In this paper, we study the numerical solution of the variational inequalities involving quasimonotone operators in infinite-dimensional Hilbert spaces. We prove that the iterative sequence generated by the proposed algorithm for the solution of quasimonotone variational inequalities converges strongly to a solution. The main advantage of the proposed iterative schemes is that it uses a monotone and non-monotone step size rule based on operator knowledge rather than its Lipschitz constant or some other line search method.

scholarly journals Two Nonmonotonic Self-Adaptive Strongly Convergent Projection-Type Methods for Solving Pseudomonotone Variational Inequalities

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Chainarong Khunpanuk ◽  
Bancha Panyanak ◽  
Nuttapol Pakkaranang
Keyword(s):  
Variational Inequalities ◽  
Iterative Methods ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Lipschitz Constant ◽  
Primary Objective ◽  
Subgradient Extragradient Method ◽  
Pseudomonotone Operator ◽  
Step Size ◽  
Iterative Schemes

The primary objective of this study is to introduce two novel extragradient-type iterative schemes for solving variational inequality problems in a real Hilbert space. The proposed iterative schemes extend the well-known subgradient extragradient method and are used to solve variational inequalities involving the pseudomonotone operator in real Hilbert spaces. The proposed iterative methods have the primary advantage of using a simple mathematical formula for step size rule based on operator information rather than the Lipschitz constant or another line search method. Strong convergence results for the suggested iterative algorithms are well-established for mild conditions, such as Lipschitz continuity and mapping monotonicity. Finally, we present many numerical experiments that show the effectiveness and superiority of iterative methods.

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 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 Improved inertial projection and contraction method for solving pseudomonotone variational inequality problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ming Tian ◽  
Gang Xu
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Lipschitz Constant ◽  
Variational Inequality Problems ◽  
Iterative Sequence ◽  
Step Size ◽  
Contraction Method ◽  
Projection And Contraction Method ◽  
Inertial Algorithm ◽  
Adaptive Step

AbstractThe objective of this article is to solve pseudomonotone variational inequality problems in a real Hilbert space. We introduce an inertial algorithm with a new self-adaptive step size rule, which is based on the projection and contraction method. Only one step projection is used to design the proposed algorithm, and the strong convergence of the iterative sequence is obtained under some appropriate conditions. The main advantage of the algorithm is that the proof of convergence of the algorithm is implemented without the prior knowledge of the Lipschitz constant of cost operator. Numerical experiments are also put forward to support the analysis of the theorem and provide comparisons with related algorithms.

scholarly journals Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications

2021 ◽  
Vol 54 (1) ◽  
pp. 280-298
Author(s):  
Nuttapol Pakkaranang ◽  
Habib ur Rehman ◽  
Wiyada Kumam
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Numerical Experiments ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Type Condition ◽  
Step Size ◽  
Extragradient Methods ◽  
Iterative Schemes ◽  
Mild Conditions

Abstract The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.

scholarly journals Inertial Iterative Schemes with Variable Step Sizes for Variational Inequality Problem Involving Pseudomonotone Operator

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 609 ◽  
Author(s):  
Jamilu Abubakar ◽  
Poom Kumam ◽  
Habib ur Rehman ◽  
Abdulkarim Hassan Ibrahim
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Variational Inequality Problem ◽  
Lipschitz Constant ◽  
Pseudomonotone Operator ◽  
Infinite Dimensional ◽  
Iterative Schemes ◽  
Computational Performance ◽  
Variable Step ◽  
Adaptive Step

Two inertial subgradient extragradient algorithms for solving variational inequality problems involving pseudomonotone operator are proposed in this article. The iterative schemes use self-adaptive step sizes which do not require the prior knowledge of the Lipschitz constant of the underlying operator. Furthermore, under mild assumptions, we show the weak and strong convergence of the sequences generated by the proposed algorithms. The strong convergence in the second algorithm follows from the use of viscosity method. Numerical experiments both in finite- and infinite-dimensional spaces are reported to illustrate the inertial effect and the computational performance of the proposed algorithms in comparison with the existing state of the art algorithms.

scholarly journals Modified Viscosity Subgradient Extragradient-Like Algorithms for Solving Monotone Variational Inequalities Problems

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 118 ◽  
Author(s):  
Nopparat Wairojjana ◽  
Mudasir Younis ◽  
Habib ur Rehman ◽  
Nuttapol Pakkaranang ◽  
Nattawut Pholasa
Keyword(s):  
Variational Inequalities ◽  
State Of The Art ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Lipschitz Constant ◽  
Mathematical Optimization ◽  
Test Problems ◽  
Monotone Variational Inequalities ◽  
Engineering Economics ◽  
Inequality Theory

Variational inequality theory is an effective tool for engineering, economics, transport and mathematical optimization. Some of the approaches used to resolve variational inequalities usually involve iterative techniques. In this article, we introduce a new modified viscosity-type extragradient method to solve monotone variational inequalities problems in real Hilbert space. The result of the strong convergence of the method is well established without the information of the operator’s Lipschitz constant. There are proper mathematical studies relating our newly designed method to the currently state of the art on several practical test problems.

scholarly journals A strong convergence theorem for solving pseudo-monotone variational inequalities and fixed point problems using subgradient extragradient method in Banach spaces

2021 ◽  
Vol 7 (4) ◽  
pp. 5015-5028
Author(s):  
Fei Ma ◽  
◽  
Jun Yang ◽  
Min Yin
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Extragradient Method ◽  
Lipschitz Constant ◽  
Fixed Point Problems ◽  
Strong Convergence Theorem ◽  
Subgradient Extragradient Method ◽  
Step Size ◽  
Common Solution

<abstract><p>In this paper, we introduce an algorithm for solving variational inequalities problem with Lipschitz continuous and pseudomonotone mapping in Banach space. We modify the subgradient extragradient method with a new and simple iterative step size, and the strong convergence to a common solution of the variational inequalities and fixed point problems is established without the knowledge of the Lipschitz constant. Finally, a numerical experiment is given in support of our results.</p></abstract>

scholarly journals A Forward-Backward-Forward Algorithm for Solving Quasimonotone Variational Inequalities

2022 ◽  
Vol 2022 ◽  
pp. 1-8
Author(s):  
Tzu-Chien Yin ◽  
Nawab Hussain
Keyword(s):  
Variational Inequalities ◽  
Convergence Analysis ◽  
Prior Knowledge ◽  
Hilbert Spaces ◽  
Lipschitz Constant ◽  
Weak Continuity ◽  
Forward Algorithm ◽  
Adaptive Technique ◽  
Quasimonotone Operators ◽  
Self Adaptive

In this paper, we continue to investigate the convergence analysis of Tseng-type forward-backward-forward algorithms for solving quasimonotone variational inequalities in Hilbert spaces. We use a self-adaptive technique to update the step sizes without prior knowledge of the Lipschitz constant of quasimonotone operators. Furthermore, we weaken the sequential weak continuity of quasimonotone operators to a weaker condition. Under some mild assumptions, we prove that Tseng-type forward-backward-forward algorithm converges weakly to a solution of quasimonotone variational inequalities.

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