scholarly journals Inertial method for split null point problems with pseudomonotone variational inequality problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chibueze Christian Okeke ◽  
Abdulmalik Usman Bello ◽  
Lateef Olakunle Jolaoso ◽  
Kingsley Chimuanya Ukandu
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Real Hilbert Space ◽  
Convergence Result ◽  
Null Point ◽  
Point Problem ◽  
Inertial Method ◽  
Convergence Results ◽  
Split Null Point Problem ◽  
Self Adaptive

<p style='text-indent:20px;'>This paper analyzed the new extragradient type algorithm with inertial extrapolation step for solving self adaptive split null point problem and pseudomonotone variational inequality in real Hilbert space. Furthermore, in this study, a strong convergence result is obtained without assuming Lipschitz continuity of the associated mapping and the operator norm is self adaptive. Additionally, the proposed algorithm only uses one projections onto the feasible set in each iteration. More so, the strong convergence results are obtained under some relaxed conditions on the initial factor and the iterative parameters. Numerical results are presented to illustrate the performance of the proposed algorithm.The results obtained in this study improved and extended related studies in the literature.</p>

scholarly journals A STRONG CONVERGENCE AND NUMERICAL ILLUSTRATION FOR THE ITERATIVE METHODS TO SOLVE A SPLIT COMMON NULL POINT PROBLEM AND A VARIATIONAL INEQUALITY IN HILBERT SPACES

2021 ◽  
Vol 226 (15) ◽  
pp. 20-27
Author(s):  
Nguyễn Thị Dinh ◽  
Phạm Thanh Hiếu
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Iterative Methods ◽  
Hilbert Spaces ◽  
Null Point ◽  
Point Problem ◽  
Numerical Illustration

Trong bài báo này, chúng tôi giới thiệu hai thuật toán lặp để giải bài toán không điểm chung tách và bài toán bất đẳng thức biến phân đơn điệu trong không gian Hilbert. Các bài toán này có nhiều ứng dụng quan trọng trong những lĩnh vực như xử lý tín hiệu, xử lý ảnh, điều khiển tối ưu và nhiều lĩnh vực khác của toán học cũng như trong đời sống. Các phương pháp mà chúng tôi đề xuất dựa trên phương pháp lặp Halper n và phương pháp xấp xỉ mềm đã được áp dụng để giải các bài toán điểm bất động và bất đẳng thức biến phân. Sự hội tụ mạnh của thuật toán đã được chứng minh cùng với một số điều kiện nhất định đặt lên các dãy tham số. Cuối cùng chúng tôi đưa ra một ví dụ số giải bài toán tối ưu trong không gian hữu hạn chiều để minh họa cho sự hội tụ của thuật toán.

scholarly journals Inertial-Like Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive and Strictly Pseudocontractive Mappings

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 860 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Adrian Petruşel ◽  
Ching-Feng Wen ◽  
Jen-Chih Yao
Keyword(s):  
Variational Inequality ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Pseudomonotone Operator ◽  
Strictly Pseudocontractive Mapping ◽  
Common Solution ◽  
Convergence Results ◽  
Asymptotically Nonexpansive

Let VIP indicate the variational inequality problem with Lipschitzian and pseudomonotone operator and let CFPP denote the common fixed-point problem of an asymptotically nonexpansive mapping and a strictly pseudocontractive mapping in a real Hilbert space. Our object in this article is to establish strong convergence results for solving the VIP and CFPP by utilizing an inertial-like gradient-like extragradient method with line-search process. Via suitable assumptions, it is shown that the sequences generated by such a method converge strongly to a common solution of the VIP and CFPP, which also solves a hierarchical variational inequality (HVI).

A Strong Convergence Theorem for Split Null Point Problem and Generalized Mixed Equilibrium Problem in Real Hilbert Spaces

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 16
Author(s):  
Olawale Kazeem Oyewole ◽  
Oluwatosin Temitope Mewomo
Keyword(s):  
Strong Convergence ◽  
Equilibrium Problem ◽  
Hilbert Spaces ◽  
Null Point ◽  
Generalized Mixed Equilibrium Problem ◽  
Point Problem ◽  
Mixed Equilibrium Problem ◽  
Generalized Mixed Equilibrium ◽  
Mixed Equilibrium ◽  
Split Null Point Problem

In this paper, we study a schematic approximation of solutions of a split null point problem for a finite family of maximal monotone operators in real Hilbert spaces. We propose an iterative algorithm that does not depend on the operator norm which solves the split null point problem and also solves a generalized mixed equilibrium problem. We prove a strong convergence of the proposed algorithm to a common solution of the two problems. We display some numerical examples to illustrate our method. Our result improves some existing results in the literature.

scholarly journals Strong Asymptotic Freeness for Free Orthogonal Quantum Groups

2014 ◽  
Vol 57 (4) ◽  
pp. 708-720 ◽  
Author(s):  
Michael Brannan
Keyword(s):  
Strong Convergence ◽  
Quantum Group ◽  
Quantum Groups ◽  
Convergence Theorem ◽  
Convergence Result ◽  
Operator Norm ◽  
Matrix Coefficient ◽  
Strong Convergence Theorem ◽  
Convergence Results ◽  
Noncommutative Polynomials

AbstractIt is known that the normalized standard generators of the free orthogonal quantum groupO+Nconverge in distribution to a free semicircular system as N → ∞. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator normof any non-commutative polynomial in the normalized standard generators ofO+Nconverges asN→ ∞ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well-knownL2-L∞norm equivalence for noncommutative polynomials in free semicircular systems.

scholarly journals Hybrid Algorithms for Variational Inequalities Involving a Strict Pseudocontraction

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1502
Author(s):  
Sun Young Cho
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Real Hilbert Space ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Strict Pseudocontraction ◽  
Adaptive Step

In a real Hilbert space, we investigate the Tseng’s extragradient algorithms with hybrid adaptive step-sizes for treating a Lipschitzian pseudomonotone variational inequality problem and a strict pseudocontraction fixed-point problem, which are symmetry. By imposing some appropriate weak assumptions on parameters, we obtain a norm solution of the problems, which solves a certain hierarchical variational inequality.

scholarly journals Strong convergence of multivariate maxima

2020 ◽  
Vol 57 (1) ◽  
pp. 314-331
Author(s):  
Michael Falk ◽  
Simone A. Padoan ◽  
Stefano Rizzelli
Keyword(s):  
Strong Convergence ◽  
Higher Dimensions ◽  
Convergence Result ◽  
Common Distribution ◽  
Convergence Results ◽  
Negative Exponential Distribution ◽  
Common Distribution Function ◽  
Variation Distance ◽  
Negative Exponential ◽  
Variational Distance

AbstractIt is well known and readily seen that the maximum of n independent and uniformly on [0, 1] distributed random variables, suitably standardised, converges in total variation distance, as n increases, to the standard negative exponential distribution. We extend this result to higher dimensions by considering copulas. We show that the strong convergence result holds for copulas that are in a differential neighbourhood of a multivariate generalised Pareto copula. Sklar’s theorem then implies convergence in variational distance of the maximum of n independent and identically distributed random vectors with arbitrary common distribution function and (under conditions on the marginals) of its appropriately normalised version. We illustrate how these convergence results can be exploited to establish the almost-sure consistency of some estimation procedures for max-stable models, using sample maxima.

scholarly journals Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Caiping Yang ◽  
Songnian He
Keyword(s):  
Variational Inequality ◽  
Unique Solution ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Monotone Operators ◽  
Strongly Monotone ◽  
Lipschitz Constants ◽  
Strongly Monotone Operators ◽  
Original Domain ◽  
Self Adaptive

Consider the variational inequalityVI(C,F)of finding a pointx*∈Csatisfying the property〈Fx*,x-x*〉≥0for allx∈C, whereCis a level set of a convex function defined on a real Hilbert spaceHandF:H→His a boundedly Lipschitzian (i.e., Lipschitzian on bounded subsets ofH) and strongly monotone operator. He and Xu proved that this variational inequality has a unique solution and devised iterative algorithms to approximate this solution (see He and Xu, 2009). In this paper, relaxed and self-adaptive iterative algorithms are proposed for computing this unique solution. Since our algorithms avoid calculating the projectionPC(calculatingPCby computing a sequence of projections onto half-spaces containing the original domainC) directly and select the stepsizes through a self-adaptive way (having no need to know any information of bounded Lipschitz constants ofF(i.e., Lipschitz constants on some bounded subsets ofH)), the implementations of our algorithms are very easy. The algorithms in this paper improve and extend the corresponding results of He and Xu.

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