scholarly journals Vector and Ordered Variational Inequalities and Applications to Order-Optimization Problems on Banach Lattices

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Linsen Xie ◽  
Jinlu Li ◽  
Wenshan Yang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Fixed Point Theorems ◽  
Optimization Problems ◽  
Banach Lattices ◽  
Variational Inequality Problems ◽  
Vector Spaces ◽  
Real Vector ◽  
Finite Dimensional

We investigate the connections between vector variational inequalities and ordered variational inequalities in finite dimensional real vector spaces. We also use some fixed point theorems to prove the solvability of ordered variational inequality problems and their application to some order-optimization problems on the Banach lattices.

scholarly journals A class of nonlinear variational inequalities involving pseudomonotone operators

2000 ◽  
Vol 13 (1) ◽  
pp. 73-75
Author(s):  
Ram U. Verma
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Monotone Operators ◽  
Variational Inequality Problems ◽  
Vector Spaces ◽  
Pseudomonotone Operators ◽  
Topological Vector Spaces ◽  
Locally Convex

We present the solvability of a class of nonlinear variational inequalities involving pseudomonotone operators in a locally convex Hausdorff topological vector spaces setting. The obtained result generalizes similar variational inequality problems on monotone operators.

scholarly journals Some new deterministic and random variational inequalities and their applications

1995 ◽  
Vol 8 (4) ◽  
pp. 381-391 ◽  
Author(s):  
Xian-Zhi Yuan ◽  
Jean-Marc Roy
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Best Approximation ◽  
Fixed Point Theorems ◽  
Approximation Theorem ◽  
Saddle Points ◽  
Vector Spaces ◽  
Random Fixed Point ◽  
Topological Vector Spaces ◽  
Random Variational Inequalities

A non-compact deterministic variational inequality which is used to prove an existence theorem for saddle points in the setting of topological vector spaces and a random variational inequality. The latter result is then applied to obtain the random version of the Fan's best approximation theorem. Several random fixed point theorems are obtained as applications of the random best approximation theorem.

scholarly journals GENERALIZED IMPLICIT INCLUSION PROBLEMS ON NONCOMPACT SETS WITH APPLICATIONS

2011 ◽  
Vol 84 (2) ◽  
pp. 261-279
Author(s):  
SAN-HUA WANG ◽  
NAN-JING HUANG
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Equilibrium Problems ◽  
Optimization Problems ◽  
Existence Results ◽  
Vector Spaces ◽  
Vector Equilibrium Problems ◽  
Inclusion Problems ◽  
Topological Vector Spaces ◽  
Vector Equilibrium

AbstractIn this paper, a class of generalized implicit inclusion problems is introduced, which can be regarded as a generalization of variational inequality problems, equilibrium problems, optimization problems and inclusion problems. Some existence results of solutions for such problems are obtained on noncompact subsets of Hausdorff topological vector spaces using the famous FKKM theorem. As applications, some existence results for vector equilibrium problems and vector variational inequalities on noncompact sets of Hausdorff topological vector spaces are given.

scholarly journals Approximate-Karush-Kuhn-Tucker Conditions and Interval Valued Vector Variational Inequalities

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Optimality Conditions ◽  
Optimization Problems ◽  
Vector Variational Inequality ◽  
Vector Variational Inequalities ◽  
Variational Inequality Problems ◽  
Objective Functions ◽  
Interval Valued

This Article deals with the Approximate Karush-Kuhn-Tucker (AKKT) optimality conditions for interval valued multiobjective function as a generalization of Karush-Kuhn-Tucker optimality conditions. Further, we establish relationship between vector variational inequality problems and multiobjective interval valued optimization problems under the assumption of LU−convex smooth and nonsmooth objective functions.

scholarly journals A new class of bilevel weak vector variational inequality problems

2020 ◽  
pp. 321-331
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Vector Variational Inequality ◽  
Variational Inequality Problems ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
New Class ◽  
Existence Conditions ◽  
Locally Convex ◽  
Weak Vector

In this paper, we first introduce a new class of bilevel weak vector variational inequality problems in locally convex Hausdorff topological vector spaces. Then, using the Kakutani-Fan-Glicksberg fixed-point theorem, we establish some existence conditions of the solution for this problem.

Topological equivalence and rigidity of flows on certain solvmanifolds

1999 ◽  
Vol 19 (3) ◽  
pp. 559-569
Author(s):  
D. BENARDETE ◽  
S. G. DANI
Keyword(s):  
Lie Group ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Real Vector ◽  
Orbit Equivalent ◽  
Finite Dimensional ◽  
Generic Class ◽  
Induced Flow ◽  
Simply Connected ◽  
Solvable Lie Group

Given a Lie group $G$ and a lattice $\Gamma$ in $G$, a one-parameter subgroup $\phi$ of $G$ is said to be rigid if for any other one-parameter subgroup $\psi$, the flows induced by $\phi$ and $\psi$ on $\Gamma\backslash G$ (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if $G$ is a simply connected solvable Lie group such that all the eigenvalues of $\mathrm{Ad} (g) $, $g\in G$, are real, then all one-parameter subgroups of $G$ are rigid for any lattice in $G$. Here we consider a complementary case, in which the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, form the unit circle of complex numbers.Let $G$ be the semidirect product $N \rtimes M$, where $M$ and $N$ are finite-dimensional real vector spaces and where the action of $M$ on the normal subgroup $N$ is such that the center of $G$ is a lattice in $M$. We prove that there is a generic class of abelian lattices $\Gamma$ in $G$ such that any semisimple one-parameter subgroup $\phi$ (namely $\phi$ such that $\mathrm{Ad} (\phi_t)$ is diagonalizable over the complex numbers for all $t$) is rigid for $\Gamma$ (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple $\phi$ are not rigid (see Corollary 4.3); further, there are non-rigid semisimple $\phi$ for which the induced flow is ergodic.

scholarly journals Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Qamrul Hasan Ansari ◽  
Chin-Tzong Pang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Hybrid Extragradient Method ◽  
Point Set ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

scholarly journals An Interior Projected-Like Subgradient Method for Mixed Variational Inequalities

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Guo-ji Tang ◽  
Xing Wang
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Subgradient Method ◽  
Mixed Variational Inequality ◽  
Convergence Estimate ◽  
Finite Dimensional ◽  
Mixed Variational Inequalities

An interior projected-like subgradient method for mixed variational inequalities is proposed in finite dimensional spaces, which is based on using non-Euclidean projection-like operator. Under suitable assumptions, we prove that the sequence generated by the proposed method converges to a solution of the mixed variational inequality. Moreover, we give the convergence estimate of the method. The results presented in this paper generalize some recent results given in the literatures.

Sign in / Sign up

Export Citation Format

Share Document