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 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 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 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.

Modified inertial subgradient extragradient method for equilibrium problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Lateef Olakunle Jolaoso ◽  
Yekini Shehu ◽  
Regina N. Nwokoye
Keyword(s):  
Variational Inequalities ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Extragradient Method ◽  
Subgradient Extragradient Method ◽  
Weak And Strong Convergence ◽  
Extragradient Methods ◽  
Convergence Results ◽  
Inertial Extrapolation ◽  
Special Case

Abstract The subgradient extragradient method with inertial extrapolation step x n + θ n (x n − x n−1) (also known as inertial subgradient extragradient method) has been studied extensively in the literature for solving variational inequalities and equilibrium problems. Most of the inertial subgradient extragradient methods in the literature for both variational inequalities and equilibrium problems have not considered the special case when the inertial factor θ n = 1. The convergence results have always been obtained when the inertial factor θ n is assumed 0 ≤ θ n < 1. This paper considers the relaxed inertial version of subgradient extragradient method for equilibrium problems with 0 ≤ θ n ≤ 1. We give both weak and strong convergence results using this inertial subgradient extragradient method and also give some numerical illustrations.

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