scholarly journals A NOTE ON A REGULARIZED GAP FUNCTION OF QVI IN BANACH SPACES

2014 ◽  
Vol 27 (2) ◽  
pp. 271-276 ◽  
Author(s):  
Sangho Kum
Keyword(s):  
Banach Spaces ◽  
Gap Function ◽  
Regularized Gap Function

scholarly journals Expected Residual Minimization Method for a Class of Stochastic Quasivariational Inequality Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Hui-Qiang Ma ◽  
Nan-Jing Huang
Keyword(s):  
Sample Average Approximation ◽  
Gap Function ◽  
Stationary Points ◽  
Quasivariational Inequality ◽  
Minimization Method ◽  
Limiting Behavior ◽  
Regularized Gap Function ◽  
Expected Residual Minimization ◽  
Sample Average ◽  
Residual Minimization

We consider the expected residual minimization method for a class of stochastic quasivariational inequality problems (SQVIP). The regularized gap function for quasivariational inequality problem (QVIP) is in general not differentiable. We first show that the regularized gap function is differentiable and convex for a class of QVIPs under some suitable conditions. Then, we reformulate SQVIP as a deterministic minimization problem that minimizes the expected residual of the regularized gap function and solve it by sample average approximation (SAA) method. Finally, we investigate the limiting behavior of the optimal solutions and stationary points.

scholarly journals On Gap Functions for Quasi-Variational Inequalities

2008 ◽  
Vol 2008 ◽  
pp. 1-7 ◽  
Author(s):  
Kouichi Taji
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Equilibrium Problems ◽  
Optimization Problems ◽  
Gap Function ◽  
Gap Functions ◽  
Merit Functions ◽  
Regularized Gap Function ◽  
Generalized Equilibrium ◽  
Quasi Variational Inequality

For variational inequalities, various merit functions, such as the gap function, the regularized gap function, the D-gap function and so on, have been proposed. These functions lead to equivalent optimization formulations and are used to optimization-based methods for solving variational inequalities. In this paper, we extend the regularized gap function and the D-gap functions for a quasi-variational inequality, which is a generalization of the variational inequality and is used to formulate generalized equilibrium problems. These extensions are shown to formulate equivalent optimization problems for quasi-variational inequalities and are shown to be continuous and directionally differentiable.

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