scholarly journals The GAP function of a parametric mixed strong vector quasivariational inequality problem

2017 ◽  
Vol 20 (K2) ◽  
pp. 126-130
Author(s):  
Dai Xuan Le ◽  
Hung Van Nguyen ◽  
Kieu Thanh Phan
Keyword(s):  
Variational Inequality ◽  
Lower Semicontinuity ◽  
Gap Function ◽  
Vector Spaces ◽  
Quasivariational Inequality ◽  
Nonlinear Scalarization Function ◽  
Nonlinear Scalarization ◽  
Inequality Problem ◽  
Topological Vector Spaces ◽  
Scalarization Function

The parametric mixed strong vector quasivariational inequality problem contains many problems such as, variational inequality problems, fixed point problems, coincidence point problems, complementary problems etc. There are many authors who have been studied the gap functions for vector variational inequality problem. This problem plays an important role in many fields of applied mathematics, especially theory of optimization. In this paper, we study a parametric gap function without the help of the nonlinear scalarization function for a parametric mixed strong vector quasivariational inequality problem (in short, (SQVIP)) in Hausdorff topological vector spaces. (SQVIP) Find x ̅ ∈ K(x ̅ ,γ) and z ̅ ∈ T(x ̅ ,γ) such that < z ̅ , y-x ̅  >+ f(y, x ̅ ,γ) ∈ Rn+ ∀ y ∈ K(x ̅ ,γ), where we denote the nonnegative of Rn by Rn+= {t=(t1 ,t2,…,tn )T ∈ Rn |ti >0, i = 1,2, ...,n}. Moreover, we also discuss the lower semicontinuity, upper semicontinuity and the continuity for the parametric gap function for this problem. To the best of our knowledge, until now there have not been any paper devoted to the lower semicontinuity, continuity of the gap function without the help of the nonlinear scalarization function for a parametric mixed strong vector quasivariational inequality problem in Hausdorff topological vector spaces. Hence the results presented in this paper (Theorem 1.3 and Theorem 1.4) are new and different in comparison with some main results in the literature.

scholarly journals ON PSEUDOMONOTONE SET-VALUED MAPPINGS IN TOPOLOGICAL VECTOR SPACES

2008 ◽  
Vol 50 (2) ◽  
pp. 258-265
Author(s):  
A. P. FARAJZADEH
Keyword(s):  
Variational Inequality ◽  
Variational Inequality Problem ◽  
Vector Spaces ◽  
Inequality Problem ◽  
Topological Vector Spaces ◽  
Set Valued Mappings

AbstractIn this paper we extend results of Inoan and Kolumban on pseudomonotone set-valued mappings to topological vector spaces. An application is made to a variational inequality problem.

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 Scalarization and well-posedness for set optimization using coradiant sets

Filomat ◽  
2019 ◽  
Vol 33 (11) ◽  
pp. 3457-3471
Author(s):  
Bin Yao ◽  
Sheng Li
Keyword(s):  
Optimization Problem ◽  
Set Optimization ◽  
Well Posedness ◽  
Nonlinear Scalarization Function ◽  
Nonlinear Scalarization ◽  
Scalar Optimization ◽  
Set Optimization Problem ◽  
Order Relations ◽  
Scalarization Function

The aim of this paper is to study scalarization and well-posedness for a set-valued optimization problem with order relations induced by a coradiant set. We introduce the notions of the set criterion solution for this problem and obtain some characterizations for these solutions by means of nonlinear scalarization. The scalarization function is a generalization of the scalarization function introduced by Khoshkhabar-amiranloo and Khorram. Moveover, we define the pointwise notions of LP well-posedness, strong DH-well-posedness and strongly B-well-posedness for the set optimization problem and characterize these properties through some scalar optimization problem based on the generalized nonlinear scalarization function respectively.

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 Nonlinear Vector Variational Inequality Problems for η-Pseudomonotone Maps

2007 ◽  
Vol 2007 ◽  
pp. 1-6
Author(s):  
A. P. Farajzadeh
Keyword(s):  
Variational Inequality ◽  
Existence Result ◽  
Complementarity Problems ◽  
Vector Variational Inequality ◽  
Variational Inequality Problems ◽  
Vector Spaces ◽  
Topological Vector Spaces ◽  
New Class ◽  
Pseudomonotone Maps ◽  
Nonlinear Vector

We consider a new class of complementarity problems for η-pseudomonotone maps and obtain an existence result for their solutions in real Hausdorff topological vector spaces. Our results extend the same previous results in this literature.

scholarly journals Constrained Weak Nash-Type Equilibrium Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
W. C. Shuai ◽  
K. L. Xiang ◽  
W. Y. Zhang
Keyword(s):  
Existence Theorem ◽  
Equilibrium Problem ◽  
Equilibrium Problems ◽  
Existence Results ◽  
Nonlinear Scalarization Function ◽  
Nonlinear Scalarization ◽  
Scalarization Function ◽  
Payoff Functions

A constrained weak Nash-type equilibrium problem with multivalued payoff functions is introduced. By virtue of a nonlinear scalarization function, some existence results are established. The results extend the corresponding one of Fu (2003). In particular, if the payoff functions are singlevalued, our existence theorem extends the main results of Fu (2003) by relaxing the assumption of convexity.

scholarly journals A Characterization ofE-Benson Proper Efficiency via Nonlinear Scalarization in Vector Optimization

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Ke Quan Zhao ◽  
Yuan Mei Xia ◽  
Hui Guo
Keyword(s):  
Vector Optimization ◽  
Optimization Problems ◽  
Proper Efficiency ◽  
Vector Optimization Problems ◽  
Nonlinear Scalarization Function ◽  
Nonlinear Scalarization ◽  
Benson Proper Efficiency ◽  
Scalarization Function

A class of vector optimization problems is considered and a characterization ofE-Benson proper efficiency is obtained by using a nonlinear scalarization function proposed by Göpfert et al. Some examples are given to illustrate the main results.

scholarly journals Variational inequality problems inH-spaces

2006 ◽  
Vol 2006 ◽  
pp. 1-18
Author(s):  
Akrur Behera ◽  
Prasanta Kumar Das
Keyword(s):  
Variational Inequality ◽  
Fixed Points ◽  
Complementarity Problem ◽  
Vector Variational Inequality ◽  
Variational Inequality Problems ◽  
Vector Spaces ◽  
Lefschetz Number ◽  
Topological Vector Spaces ◽  
Invex Set ◽  
Invex Function

The concept ofη-invex set is explored and the concept ofT-η-invex function is introduced. These concepts are applied to the generalized vector variational inequality problems in ordered topological vector spaces. The study of variational inequality problems is extended toH-spaces and differentiablen-manifolds. The solution of complementarity problem is also studied in the presence of fixed points or Lefschetz number.

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