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.

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 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 Approximation Results for Equilibrium Problems Involving Strongly Pseudomonotone Bifunction in Real Hilbert Spaces

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 137
Author(s):  
Wiyada Kumam ◽  
Kanikar Muangchoo
Keyword(s):  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Minimax Problems ◽  
Strong Convergence Theorem ◽  
Test Problems ◽  
Subgradient Extragradient Method ◽  
Step Size ◽  
Pseudomonotone Bifunction ◽  
Equilibrium Test

A plethora of applications in non-linear analysis, including minimax problems, mathematical programming, the fixed-point problems, saddle-point problems, penalization and complementary problems, may be framed as a problem of equilibrium. Most of the methods used to solve equilibrium problems involve iterative methods, which is why the aim of this article is to establish a new iterative method by incorporating an inertial term with a subgradient extragradient method to solve the problem of equilibrium, which includes a bifunction that is strongly pseudomonotone and meets the Lipschitz-type condition in a real Hilbert space. Under certain mild conditions, a strong convergence theorem is proved, and a required sequence is generated without the information of the Lipschitz-type cost bifunction constants. Thus, the method operates with the help of a slow-converging step size sequence. In numerical analysis, we consider various equilibrium test problems to validate our proposed results.

scholarly journals Modified subgradient extragradient method for system of variational inclusion problem and finite family of variational inequalities problem in real Hilbert space

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Araya Kheawborisut ◽  
Atid Kangtunyakarn
Keyword(s):  
Hilbert Space ◽  
Variational Inequalities ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Finite Family ◽  
Variational Inclusion ◽  
Common Element ◽  
Inclusion Problem ◽  
Subgradient Extragradient Method ◽  
Generalized System

AbstractFor the purpose of this article, we introduce a modified form of a generalized system of variational inclusions, called the generalized system of modified variational inclusion problems (GSMVIP). This problem reduces to the classical variational inclusion and variational inequalities problems. Motivated by several recent results related to the subgradient extragradient method, we propose a new subgradient extragradient method for finding a common element of the set of solutions of GSMVIP and the set of a finite family of variational inequalities problems. Under suitable assumptions, strong convergence theorems have been proved in the framework of a Hilbert space. In addition, some numerical results indicate that the proposed method is effective.

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 Mildly Inertial Subgradient Extragradient Method for Variational Inequalities Involving an Asymptotically Nonexpansive and Finitely Many Nonexpansive Mappings

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 881 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Xiaolong Qin ◽  
Yekini Shehu ◽  
Jen-Chih Yao
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Subgradient Extragradient Method ◽  
Pseudomonotone Operator ◽  
Common Solution ◽  
Asymptotically Nonexpansive

In a real Hilbert space, let the notation VIP indicate a variational inequality problem for a Lipschitzian, pseudomonotone operator, and let CFPP denote a common fixed-point problem of an asymptotically nonexpansive mapping and finitely many nonexpansive mappings. This paper introduces mildly inertial algorithms with linesearch process for finding a common solution of the VIP and the CFPP by using a subgradient approach. These fully absorb hybrid steepest-descent ideas, viscosity iteration ideas, and composite Mann-type iterative ideas. With suitable conditions on real parameters, it is shown that the sequences generated our algorithms converge to a common solution in norm, which is a unique solution of a hierarchical variational inequality (HVI).

scholarly journals Analysis of Subgradient Extragradient Iterative Schemes for Variational Inequalities

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Danfeng Wu ◽  
Li-Jun Zhu ◽  
Zhuang Shan ◽  
Tzu-Chien Yin
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Convergence Analysis ◽  
Hilbert Spaces ◽  
Extragradient Method ◽  
Subgradient Extragradient Method ◽  
Iterative Schemes ◽  
Mild Conditions ◽  
Monotone Variational Inequality

In this paper, we investigate the monotone variational inequality in Hilbert spaces. Based on Censor’s subgradient extragradient method, we propose two modified subgradient extragradient algorithms with self-adaptive and inertial techniques for finding the solution of the monotone variational inequality in real Hilbert spaces. Strong convergence analysis of the proposed algorithms have been obtained under some mild conditions.

scholarly journals Halpern-type relaxed inertial algorithms with Bregman divergence for solving variational inequalities

2021 ◽  
Author(s):  
Lateef Olakunle Jolaoso ◽  
Pongsakorn Sunthrayuth ◽  
Prasit Cholamjiak ◽  
Yeol Je Cho
Keyword(s):  
Applied Sciences ◽  
Variational Inequalities ◽  
Lipschitz Continuity ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Weak Continuity ◽  
Subgradient Extragradient Method ◽  
Bregman Divergence ◽  
Effective Technique ◽  
Tseng’S Extragradient Method

Abstract It is well-known that the use of Bregman divergence is an elegant and effective technique for solving many problems in applied sciences. In this paper, we introduce and analyze two new inertial-like algorithms with Bregman divergence for solving pseudomonotone variational inequalities in a real Hilbert space. The first algorithm is inspired by Halpern -type iteration and subgradient extragradient method and the second algorithm is inspired by Halpern -type iteration and Tseng's extragradient method. Under suitable conditions, the strong convergence theorems of the algorithms are established without assuming the Lipschitz continuity and the sequential weak continuity of any mapping. Finally, several numerical experiments with various types of Bregman divergence are also performed to illustrate the theoretical analysis. The results presented in this paper improve and generalize the related works in the literature.

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.

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