## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2013, Special Issue (2013), Article ID 423040, 8 pages.

### A New Gap Function for Vector Variational Inequalities with an Application

#### 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.

#### Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2013), Article ID 423040, 8 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394807743

Digital Object Identifier
doi:10.1155/2013/423040

Mathematical Reviews number (MathSciNet)
MR3068875

Zentralblatt MATH identifier
1271.49004

#### Citation

Ma, Hui-qiang; Huang, Nan-jing; Wu, Meng; O'Regan, Donal. A New Gap Function for Vector Variational Inequalities with an Application. J. Appl. Math. 2013, Special Issue (2013), Article ID 423040, 8 pages. doi:10.1155/2013/423040. https://projecteuclid.org/euclid.jam/1394807743

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