scholarly journals Strong pseudomonotonicity, sharp efficiency and stability for parametric vector equilibria

2007 ◽  
Vol 17 ◽  
pp. 9-18 ◽  
Author(s):  
Ewa M. Bednarczuk
Keyword(s):  
Strong Pseudomonotonicity

scholarly journals Continuity Results and Error Bounds on Pseudomonotone Vector Variational Inequalities via Scalarization

2016 ◽  
Vol 2016 ◽  
pp. 1-11
Author(s):  
Xin Zuo ◽  
Hong-Zhi Wei ◽  
Chun-Rong Chen
Keyword(s):  
Variational Inequality ◽  
Error Bounds ◽  
Vector Variational Inequality ◽  
Gap Functions ◽  
Strong Pseudomonotonicity ◽  
Solution Mapping ◽  
Constraint Set ◽  
Regularized Gap Functions ◽  
Lower And Upper Semicontinuities ◽  
Weak Vector

Continuity (both lower and upper semicontinuities) results of the Pareto/efficient solution mapping for a parametric vector variational inequality with a polyhedral constraint set are established via scalarization approaches, within the framework of strict pseudomonotonicity assumptions. As a direct application, the continuity of the solution mapping to a parametric weak Minty vector variational inequality is also discussed. Furthermore, error bounds for the weak vector variational inequality in terms of two known regularized gap functions are also obtained, under strong pseudomonotonicity assumptions.

scholarly journals Existence results and two step proximal point algorithm for equilibrium problems on Hadamard manifolds

2021 ◽  
Vol 37 (3) ◽  
pp. 393-406
Author(s):  
SULIMAN AL-HOMIDAN ◽  
◽  
QAMRUL HASAN ANSARI ◽  
MONIRUL ISLAM ◽  
◽  
...  
Keyword(s):  
Proximal Point Algorithm ◽  
Equilibrium Problems ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Coercivity Conditions ◽  
Hadamard Manifolds ◽  
Type Condition ◽  
Proximal Point ◽  
Strong Pseudomonotonicity

"In this paper, we study the existence of solutions of equilibrium problems in the setting of Hadamard manifolds under the pseudomonotonicity and geodesic upper sign continuity of the equilibrium bifunction and under different kinds of coercivity conditions. We also study the existence of solutions of the equilibrium problems under properly quasimonotonicity of the equilibrium bifunction. We propose a two-step proximal point algorithm for solving equilibrium problems in the setting of Hadamard manifolds. The convergence of the proposed algorithm is studied under the strong pseudomonotonicity and Lipschitz-type condition. The results of this paper either extend or generalize several known results in the literature."

scholarly journals Variational-Like Inequalities and Equilibrium Problems with Generalized Monotonicity in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
N. K. Mahato ◽  
C. Nahak
Keyword(s):  
Equilibrium Problem ◽  
Banach Spaces ◽  
Equilibrium Problems ◽  
Existence Of Solutions ◽  
Existence Of Solution ◽  
Generalized Monotonicity ◽  
Reflexive Banach Spaces ◽  
Strong Pseudomonotonicity ◽  
Pseudomonotone Maps ◽  
Pseudomonotone Mappings

We introduce the notion of relaxed (ρ-θ)-η-invariant pseudomonotone mappings, which is weaker than invariant pseudomonotone maps. Using the KKM technique, we establish the existence of solutions for variational-like inequality problems with relaxed (ρ-θ)-η-invariant pseudomonotone mappings in reflexive Banach spaces. We also introduce the concept of (ρ-θ)-pseudomonotonicity for bifunctions, and we consider some examples to show that (ρ-θ)-pseudomonotonicity generalizes both monotonicity and strong pseudomonotonicity. The existence of solution for equilibrium problem with (ρ-θ)-pseudomonotone mappings in reflexive Banach spaces are demonstrated by using the KKM technique.

An inertial modified algorithm for solving variational inequalities

2020 ◽  
Vol 54 (1) ◽  
pp. 163-178 ◽  
Author(s):  
Dang Van Hieu ◽  
Pham Kim Quy
Keyword(s):  
Hilbert Spaces ◽  
Lipschitz Constant ◽  
Inertial Effects ◽  
Convergence Properties ◽  
Pseudomonotone Operators ◽  
Lipschitz Continuous ◽  
Strong Pseudomonotonicity ◽  
Computational Performance ◽  
Speed Up ◽  
The Cost

The paper deals with an inertial-like algorithm for solving a class of variational inequality problems involving Lipschitz continuous and strongly pseudomonotone operators in Hilbert spaces. The presented algorithm can be considered a combination of the modified subgradient extragradient-like algorithm and inertial effects. This is intended to speed up the convergence properties of the algorithm. The main feature of the new algorithm is that it is done without the prior knowledge of the Lipschitz constant and the modulus of strong pseudomonotonicity of the cost operator. Several experiments are performed to illustrate the convergence and computational performance of the new algorithm in comparison with others having similar features. The numerical results have confirmed that the proposed algorithm has a competitive advantage over the existing methods.

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