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.

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 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 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.

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 An Efficient Parallel Extragradient Method for Systems of Variational Inequalities Involving Fixed Points of Demicontractive Mappings

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1915
Author(s):  
Lateef Olakunle Jolaoso ◽  
Maggie Aphane
Keyword(s):  
Variational Inequalities ◽  
Hilbert Spaces ◽  
Line Search ◽  
Convex Combination ◽  
Extragradient Method ◽  
Convergence Result ◽  
Armijo Line Search ◽  
Systems Of Variational Inequalities ◽  
Demicontractive Mappings ◽  
The Cost

Herein, we present a new parallel extragradient method for solving systems of variational inequalities and common fixed point problems for demicontractive mappings in real Hilbert spaces. The algorithm determines the next iterate by computing a computationally inexpensive projection onto a sub-level set which is constructed using a convex combination of finite functions and an Armijo line-search procedure. A strong convergence result is proved without the need for the assumption of Lipschitz continuity on the cost operators of the variational inequalities. Finally, some numerical experiments are performed to illustrate the performance of the proposed method.

Relaxed extragradient algorithm for solving pseudomonotone variational inequalities in Hilbert spaces

Optimization ◽  
2019 ◽  
Vol 69 (10) ◽  
pp. 2279-2304 ◽  
Author(s):  
Dang Van Hieu ◽  
Yeol Je Cho ◽  
Yi-bin Xiao ◽  
Poom Kumam
Keyword(s):  
Variational Inequalities ◽  
Hilbert Spaces ◽  
Extragradient Algorithm

scholarly journals Strong Convergence of a Unified General Iteration fork-Strictly Pseudononspreading Mapping in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Dao-Jun Wen ◽  
Yi-An Chen ◽  
Yan Tang
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Minimization Problem ◽  
Iterative Sequence ◽  
Mean Convergence ◽  
Strictly Pseudononspreading Mapping ◽  
General Iterative Method

We introduce a unified general iterative method to approximate a fixed point ofk-strictly pseudononspreading mapping. Under some suitable conditions, we prove that the iterative sequence generated by the proposed method converges strongly to a fixed point of ak-strictly pseudononspreading mapping with an idea of mean convergence, which also solves a class of variational inequalities as an optimality condition for a minimization problem. The results presented in this paper may be viewed as a refinement and as important generalizations of the previously known results announced by many other authors.

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.

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