scholarly journals Viscosity method for hierarchical variational inequalities and variational inclusions on Hadamard manifolds

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Doaa Filali ◽  
Mohammad Dilshad ◽  
Mohammad Akram ◽  
Feeroz Babu ◽  
Izhar Ahmad
Keyword(s):  
Variational Inequalities ◽  
Optimization Problem ◽  
Contraction Mapping ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
Viscosity Method ◽  
Common Solution ◽  
Convergence Results ◽  
Hierarchical Variational Inequalities ◽  
Nonsmooth Optimization Problem

AbstractThis article aims to introduce and analyze the viscosity method for hierarchical variational inequalities involving a ϕ-contraction mapping defined over a common solution set of variational inclusion and fixed points of a nonexpansive mapping on Hadamard manifolds. Several consequences of the composed method and its convergence theorem are presented. The convergence results of this article generalize and extend some existing results from Hilbert/Banach spaces and from Hadamard manifolds. We also present an application to a nonsmooth optimization problem. Finally, we clarify the convergence analysis of the proposed method by some computational numerical experiments in Hadamard manifold.

scholarly journals Viscosity method with a φ-contraction mapping for hierarchical variational inequalities on Hadamard manifolds

2020 ◽  
Vol 21 (2) ◽  
pp. 561-584
Author(s):  
Suliman Al-Homidan ◽  
◽  
Qamrul Hasan Ansari ◽  
Feeroz Babu ◽  
Jen-Chih Yao ◽  
...  
Keyword(s):  
Variational Inequalities ◽  
Contraction Mapping ◽  
Hadamard Manifolds ◽  
Viscosity Method ◽  
Hierarchical Variational Inequalities

scholarly journals Solving Yosida inclusion problem in Hadamard manifold

2019 ◽  
Vol 9 (2) ◽  
pp. 357-366 ◽  
Author(s):  
Mohammad Dilshad
Keyword(s):  
Approximate Solution ◽  
Vector Field ◽  
Variational Inequalities ◽  
Hadamard Manifold ◽  
Inclusion Problem ◽  
Approximation Operator ◽  
Hadamard Manifolds ◽  
Yosida Approximation ◽  
Type Algorithm ◽  
Monotone Vector Field

Abstract We consider a Yosida inclusion problem in the setting of Hadamard manifolds. We study Korpelevich-type algorithm for computing the approximate solution of Yosida inclusion problem. The resolvent and Yosida approximation operator of a monotone vector field and their properties are used to prove that the sequence generated by the proposed algorithm converges to the solution of Yosida inclusion problem. An application to our problem and algorithm is presented to solve variational inequalities in Hadamard manifolds.

scholarly journals Explicit Iterative Method for Variational Inequalities on Hadamard Manifolds

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Iterative Method ◽  
Variational Inequalities ◽  
New Method ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
Auxiliary Principle ◽  
Auxiliary Principle Technique ◽  
Monotonicity Results

An explicit iterative method for solving the variational inequalities on Hadamard manifold is suggested and analyzed using the auxiliary principle technique. The convergence of this new method requires only the partially relaxed strongly monotonicity, which is a weaker condition than monotonicity. Results can be viewed as refinement and improvement of previously known results.

scholarly journals Proximal Point Methods for Solving Mixed Variational Inequalities on the Hadamard Manifolds

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Variational Inequalities ◽  
Point Method ◽  
Hadamard Manifold ◽  
Proximal Point Method ◽  
Hadamard Manifolds ◽  
Auxiliary Principle ◽  
Proximal Point ◽  
Auxiliary Principle Technique ◽  
Special Cases ◽  
Mixed Variational Inequalities

We use the auxiliary principle technique to suggest and analyze a proximal point method for solving the mixed variational inequalities on the Hadamard manifold. It is shown that the convergence of this proximal point method needs only pseudomonotonicity, which is a weaker condition than monotonicity. Some special cases are also considered. Results can be viewed as refinement and improvement of previously known results.

scholarly journals Roughly geodesic B - r-preinvex functions on Cartan Hadamard manifolds

2018 ◽  
Vol 33 (2) ◽  
pp. 325
Author(s):  
Meraj Ali Khan ◽  
Izhar Ahmad
Keyword(s):  
Optimization Problem ◽  
Global Minimum ◽  
Minimum Point ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
New Class ◽  
Local Minimum Point ◽  
Preinvex Functions ◽  
Global Minimum Point ◽  
Class Of Functions

In this article, we introduce a new class of functions called roughly geodesic B????r????preinvex on a Hadamard manifold and establish some properties of roughly geodesic B - r-preinvex functions on Hadamard manifolds. It is observed that a local minimum point for a scalar optimization problem is also a global minimum point under roughly geodesic B-r- preinvexity on Hadamard manifolds. The results presented in this paper extend and generalize the results appeared in the literature.

scholarly journals On the Parallel Subgradient Extragradient Rule for Solving Systems of Variational Inequalities in Hadamard Manifolds

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1496
Author(s):  
Chun-Yan Wang ◽  
Lu-Chuan Ceng ◽  
Long He ◽  
Hui-Ying Hu ◽  
Tu-Yan Zhao ◽  
...  
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Parallel Algorithms ◽  
Vector Fields ◽  
Variational Inequality Problem ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
Inequality Problem ◽  
Symmetric Structure ◽  
Systems Of Variational Inequalities

In a Hadamard manifold, let the VIP and SVI represent a variational inequality problem and a system of variational inequalities, respectively, where the SVI consists of two variational inequalities which are of symmetric structure mutually. This article designs two parallel algorithms to solve the SVI via the subgradient extragradient approach, where each algorithm consists of two parts which are of symmetric structure mutually. It is proven that, if the underlying vector fields are of monotonicity, then the sequences constructed by these algorithms converge to a solution of the SVI. We also discuss applications of these algorithms for approximating solutions to the VIP. Our theorems complement some recent and important ones in the literature.

scholarly journals Common Solution for a Finite Family of Equilibrium Problems, Quasi-Variational Inclusion Problems and Fixed Points on Hadamard Manifolds

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1161
Author(s):  
Jinhua Zhu ◽  
Jinfang Tang ◽  
Shih-sen Chang ◽  
Min Liu ◽  
Liangcai Zhao
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Fixed Point Problem ◽  
Finite Family ◽  
Variational Inclusion ◽  
Point Problem ◽  
Hadamard Manifolds ◽  
Inclusion Problems ◽  
Common Solution

In this paper, we introduce an iterative algorithm for finding a common solution of a finite family of the equilibrium problems, quasi-variational inclusion problems and fixed point problem on Hadamard manifolds. Under suitable conditions, some strong convergence theorems are proved. Our results extend some recent results in literature.

scholarly journals Vector variational inequalities and their relations with vector optimization

2013 ◽  
Vol 4 (1) ◽  
pp. 35-44 ◽  
Author(s):  
Surjeet Kaur Suneja ◽  
Bhawna Kohli
Keyword(s):  
Variational Inequalities ◽  
Vector Optimization ◽  
Optimization Problem ◽  
Vector Optimization Problem ◽  
Clarke Subdifferential ◽  
Vector Variational Inequalities ◽  
Regularity Assumption ◽  
Weak Minimum ◽  
Minimum Solution ◽  
Clarke Subdifferentials

In this paper, K- quasiconvex, K- pseudoconvex and other related functions have been introduced in terms of their Clarke subdifferentials, where   is an arbitrary closed convex, pointed cone with nonempty interior. The (strict, weakly) -pseudomonotonicity, (strict) K- naturally quasimonotonicity and K- quasimonotonicity of Clarke subdifferential maps have also been defined. Further, we introduce Minty weak (MVVIP) and Stampacchia weak (SVVIP) vector variational inequalities over arbitrary cones. Under regularity assumption, we have proved that a weak minimum solution of vector optimization problem (VOP) is a solution of (SVVIP) and under the condition of K- pseudoconvexity we have obtained the converse for MVVIP (SVVIP). In the end we study the interrelations between these with the help of strict K-naturally quasimonotonicity of Clarke subdifferential map.

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