scholarly journals Extension of Extragradient Techniques for Variational Inequalities

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 111
Author(s):  
Yonghong Yao ◽  
Ke Wang ◽  
Xiaowei Qin ◽  
Li-Jun Zhu
Keyword(s):  
Variational Inequalities ◽  
Convergence Result ◽  
Type Method ◽  
Mild Conditions ◽  
The Common

An extragradient type method for finding the common solutions of two variational inequalities has been proposed. The convergence result of the algorithm is given under mild conditions on the algorithm parameters.

scholarly journals Iterative Algorithms for General Multivalued Variational Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Yonghong Yao ◽  
Muhammad Aslam Noor ◽  
Yeong-Cheng Liou ◽  
Shin Min Kang
Keyword(s):  
Variational Inequalities ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Convergence Result ◽  
Common Element ◽  
Projection Technique ◽  
Mild Conditions ◽  
Special Cases ◽  
The Common ◽  
Multivalued Variational Inequalities

We introduce and study some new classes of variational inequalities and the Wiener-Hopf equations. Using essentially the projection technique, we establish the equivalence between these problems. This equivalence is used to suggest and analyze some iterative methods for solving the general multivalued variational in equalities in conjunction with nonexpansive mappings. We prove a strong convergence result for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general multivalued variational inequalities under some mild conditions. Several special cases are also discussed.

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.

scholarly journals On an Aitken-Steffensen-Newton type method

2018 ◽  
Vol 34 (1) ◽  
pp. 85-92
Author(s):  
ION PAVALOIU ◽  
Keyword(s):  
Local Convergence ◽  
Convergence Result ◽  
Convergence Order ◽  
Monotone Convergence ◽  
Type Method ◽  
Numerical Examples ◽  
Newton Iterations ◽  
Convergence Orders

We consider an Aitken-Steffensen type method in which the nodes are controlled by Newton and two-step Newton iterations. We prove a local convergence result showing the q-convergence order 7 of the iterations. Under certain supplementary conditions, we obtain monotone convergence of the iterations, providing an alternative to the usual ball attraction theorems. Numerical examples show that this method may, in some cases, have larger (possibly sided) convergence domains than other methods with similar convergence orders.

scholarly journals A System of Differential Set-Valued Variational Inequalities in Finite Dimensional Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Wei Li ◽  
Xing Wang ◽  
Nan-Jing Huang
Keyword(s):  
Variational Inequalities ◽  
Existence Theorem ◽  
Weak Solutions ◽  
Convergence Result ◽  
Time Dependent ◽  
Euclidean Spaces ◽  
Finite Dimensional ◽  
Differential Set

A system of differential set-valued variational inequalities is introduced and studied in finite dimensional Euclidean spaces. An existence theorem of weak solutions for the system of differential set-valued variational inequalities in the sense of Carathéodory is proved under some suitable conditions. Furthermore, a convergence result on Euler time-dependent procedure for solving the system of differential set-valued variational inequalities is also given.

scholarly journals An Improved Two-Step Method for Generalized Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Haibin Chen
Keyword(s):  
Variational Inequalities ◽  
Error Bound ◽  
Point Of View ◽  
Large Decrease ◽  
Step Method ◽  
Generalized Variational Inequalities ◽  
Mild Conditions ◽  
Extragradient Algorithm ◽  
Globally Convergent ◽  
Geometric Point

We propose an improved two-step extragradient algorithm for pseudomonotone generalized variational inequalities. It requires two projections at each iteration and allows one to take different stepsize rules. Moreover, from a geometric point of view, it is shown that the new method has a long stepsize, and it guarantees that the distance from the next iterative point to the solution set has a large decrease. Under mild conditions, we show that the method is globally convergent, and then the R-linearly convergent property of the method is proven if a projection-type error bound holds locally.

scholarly journals Maps Preserving Norms of Generalized Weighted Quasi-arithmetic Means of Invertible Positive Operators

2019 ◽  
Vol 35 ◽  
pp. 357-364
Author(s):  
Gergő Nagy ◽  
Patricia Szokol
Keyword(s):  
General Setting ◽  
Interesting Property ◽  
Positive Operators ◽  
Arithmetic Means ◽  
Mild Conditions ◽  
Weighted Arithmetic ◽  
The Common ◽  
Adjoint Operators

In this paper, the problem of describing the structure of transformations leaving norms of generalized weighted quasi-arithmetic means of invertible positive operators invariant is discussed. In a former result of the authors, this problem was solved for weighted quasi-arithmetic means, and here the corresponding result is generalized by establishing its solution under certain mild conditions. It is proved that in a quite general setting, generalized weighted quasi-arithmetic means on self-adjoint operators are not monotone in their variables which is an interesting property. Moreover, the relation of these means with the Kubo-Ando means is investigated and it is shown that the common members of the classes of these types of means are weighted arithmetic means.

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