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 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 Strong convergence inertial projection and contraction method with self adaptive stepsize for pseudomonotone variational inequalities and fixed point problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Lateef Olakunle Jolaoso ◽  
Maggie Aphane
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Lipschitz Constant ◽  
Convergence Result ◽  
Common Element ◽  
Adaptive Technique ◽  
Prior Estimate ◽  
Adaptive Stepsize ◽  
Self Adaptive

AbstractIn this paper, we introduce a new inertial self-adaptive projection method for finding a common element in the set of solution of pseudomonotone variational inequality problem and set of fixed point of a pseudocontractive mapping in real Hilbert spaces. The self-adaptive technique ensures the convergence of the algorithm without any prior estimate of the Lipschitz constant. With the aid of Moudafi’s viscosity approximation method, we prove a strong convergence result for the sequence generated by our algorithm under some mild conditions. We also provide some numerical examples to illustrate the accuracy and efficiency of the algorithm by comparing with other recent methods in the literature.

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 Tseng-Type Algorithm with Self-Adaptive Techniques for Solving the Split Problem of Fixed Points and Pseudomonotone Variational Inequalities in Hilbert Spaces

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 152
Author(s):  
Li-Jun Zhu ◽  
Yeong-Cheng Liou
Keyword(s):  
Variational Inequalities ◽  
Fixed Points ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Pseudomonotone Operators ◽  
Adaptive Techniques ◽  
Type Algorithm ◽  
Self Adaptive

In this paper, we survey the split problem of fixed points of two pseudocontractive operators and variational inequalities of two pseudomonotone operators in Hilbert spaces. We present a Tseng-type iterative algorithm for solving the split problem by using self-adaptive techniques. Under certain assumptions, we show that the proposed algorithm converges weakly to a solution of the split problem. An application is included.

scholarly journals Iterative Algorithms for Pseudomonotone Variational Inequalities and Fixed Point Problems of Pseudocontractive Operators

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1189 ◽  
Author(s):  
Yonghong Yao ◽  
Mihai Postolache ◽  
Jen-Chih Yao
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Convergence Analysis ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Iterative Algorithms ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Common Solution

In this paper, we are interested in the pseudomonotone variational inequalities and fixed point problem of pseudocontractive operators in Hilbert spaces. An iterative algorithm has been constructed for finding a common solution of the pseudomonotone variational inequalities and fixed point of pseudocontractive operators. Strong convergence analysis of the proposed procedure is given. Several related corollaries are included.

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