Journal of Applied Mathematics
Volume 2013 (2013), Article ID 423040, 8 pages
http://dx.doi.org/10.1155/2013/423040
Research Article
A New Gap Function for Vector Variational Inequalities with an Application
1School of Economics, Southwest University for Nationalities, Chengdu, Sichuan 610041, China
2Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
3Business School, Sichuan University, Chengdu, Sichuan 610064, China
4School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
Received 29 May 2013; Accepted 4 June 2013
Academic Editor: Gue Myung Lee
Copyright © 2013 Hui-qiang Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We consider a vector variational inequality in a finite-dimensional space. A new gap function is proposed, and an equivalent optimization problem for the vector variational inequality is also provided. Under some suitable conditions, we prove that the gap function is directionally differentiable and that any point satisfying the first-order necessary optimality condition for the equivalent optimization problem solves the vector variational inequality. As an application, we use the new gap function to reformulate a stochastic vector variational inequality as a deterministic optimization problem. We solve this optimization problem by employing the sample average approximation method. The convergence of optimal solutions of the approximation problems is also investigated.